WEBVTT 00:00:16.559 --> 00:00:20.719 I couldn't connect but 00:00:18.280 --> 00:00:22.120 so the Fed just hiked by 25 basis 00:00:20.719 --> 00:00:23.160 points. 00:00:22.120 --> 00:00:25.720 And 00:00:23.160 --> 00:00:27.039 as people expected, you know, this is 00:00:25.719 --> 00:00:29.439 the way that it works when there's lots 00:00:27.039 --> 00:00:31.640 of uncertainty essentially 00:00:29.440 --> 00:00:33.160 the Fed starts communicating 00:00:31.640 --> 00:00:34.759 what's going to do 00:00:33.159 --> 00:00:36.199 and the communication was still very 00:00:34.759 --> 00:00:39.239 clear that 00:00:36.200 --> 00:00:41.359 that 25 basis points was 00:00:39.240 --> 00:00:42.719 to be expected and and apparently I was 00:00:41.359 --> 00:00:43.479 reading this right now. It was released 00:00:42.719 --> 00:00:45.439 at 00:00:43.479 --> 00:00:47.039 3 minutes ago, 4 minutes ago. 00:00:45.439 --> 00:00:50.599 Um 00:00:47.039 --> 00:00:52.679 they also said that that further hikes 00:00:50.600 --> 00:00:55.719 are no longer guaranteed. So remember 00:00:52.679 --> 00:00:57.640 that we saw that expected 00:00:55.719 --> 00:00:59.799 hikes sort of we saw several several 00:00:57.640 --> 00:01:01.119 expected hikes for the next few months 00:00:59.799 --> 00:01:05.000 before 00:01:01.119 --> 00:01:07.039 the SVB mess and right after it we sort 00:01:05.000 --> 00:01:09.599 of saw the whole thing declining and and 00:01:07.040 --> 00:01:13.160 at least the minutes are consistent with 00:01:09.599 --> 00:01:14.640 that. Um so there we are. So not big 00:01:13.159 --> 00:01:16.079 uncertainty I mean the markets are 00:01:14.640 --> 00:01:18.760 rallying or something like that at least 00:01:16.079 --> 00:01:20.879 for the next 10 minutes or so but uh 00:01:18.760 --> 00:01:23.320 we shall see. 00:01:20.879 --> 00:01:24.719 Anyway, so but today we we're going to 00:01:23.319 --> 00:01:26.000 really start 00:01:24.719 --> 00:01:28.480 I'm going to I'm going to show you sort 00:01:26.000 --> 00:01:29.840 of the first model of economic growth. 00:01:28.480 --> 00:01:31.200 Uh 00:01:29.840 --> 00:01:33.159 And 00:01:31.200 --> 00:01:35.840 before I do that, who knows who that 00:01:33.159 --> 00:01:35.840 person is? 00:01:37.680 --> 00:01:40.920 No? No clue? 00:01:41.879 --> 00:01:46.039 He 00:01:42.760 --> 00:01:48.439 actually he's Robert Solow. He was an 00:01:46.040 --> 00:01:50.080 He's an emeritus professor at MIT. 00:01:48.439 --> 00:01:52.000 Together with Paul Samuelson essentially 00:01:50.079 --> 00:01:54.519 he's responsible for building the 00:01:52.000 --> 00:01:57.079 economics department at MIT. And he won 00:01:54.519 --> 00:02:00.640 the Nobel Prize in 1987. 00:01:57.079 --> 00:02:02.519 I was a student then here. Uh and 00:02:00.640 --> 00:02:05.280 and 00:02:02.519 --> 00:02:08.079 for his work primarily for his work on 00:02:05.280 --> 00:02:10.319 economic growth. And so what we're going 00:02:08.080 --> 00:02:13.800 to do in the next two three lectures are 00:02:10.319 --> 00:02:17.799 essentially things that Bob Solow 00:02:13.800 --> 00:02:20.040 developed many many years ago. 00:02:17.800 --> 00:02:21.880 The basic mechanism, you know, remember 00:02:20.039 --> 00:02:23.319 that we had this Keynesian cross before 00:02:21.879 --> 00:02:25.199 where we have this multiplier in the 00:02:23.319 --> 00:02:26.560 goods market and aggregate demand 00:02:25.199 --> 00:02:28.079 feeding into income and so on and so 00:02:26.560 --> 00:02:31.599 forth. That was sort of the start 00:02:28.080 --> 00:02:34.960 mechanism in in in short-run macro. 00:02:31.599 --> 00:02:37.319 In long-run macro growth theory 00:02:34.960 --> 00:02:38.680 this is sort of the the key mechanism 00:02:37.319 --> 00:02:41.000 and and you can think of it as the 00:02:38.680 --> 00:02:43.120 following. At any point in time 00:02:41.000 --> 00:02:45.680 an economy has, you know, 00:02:43.120 --> 00:02:47.280 factors of production primarily labor 00:02:45.680 --> 00:02:49.200 and capital. 00:02:47.280 --> 00:02:50.960 That capital stock, labor is more or 00:02:49.199 --> 00:02:52.079 less fixed or depends on population 00:02:50.960 --> 00:02:54.879 growth, things that are sort of 00:02:52.080 --> 00:02:57.960 difficult to to control or they're not 00:02:54.879 --> 00:03:00.879 really that endogenous to to economics. 00:02:57.960 --> 00:03:02.879 Not at least in the current times. Many 00:03:00.879 --> 00:03:04.319 centuries ago yes they were. We had this 00:03:02.879 --> 00:03:06.560 Malthusian theories in which you know 00:03:04.319 --> 00:03:08.519 population growth determined determined 00:03:06.560 --> 00:03:11.039 growth because 00:03:08.520 --> 00:03:13.560 food is scarcity and stuff like that but 00:03:11.039 --> 00:03:16.239 that's no longer the case fortunately. 00:03:13.560 --> 00:03:19.479 Uh for in most parts of the world. So 00:03:16.240 --> 00:03:21.120 but what can what can change over time 00:03:19.479 --> 00:03:23.919 and quite a bit and it depends on 00:03:21.120 --> 00:03:25.000 economic decisions is the capital stock. 00:03:23.919 --> 00:03:27.079 But at any point in time there is 00:03:25.000 --> 00:03:29.520 certain capital stocks which combine 00:03:27.080 --> 00:03:31.440 with labor give you some certain output. 00:03:29.520 --> 00:03:33.480 Output is income. 00:03:31.439 --> 00:03:34.560 Part of that income will be saved as 00:03:33.479 --> 00:03:37.719 we're seeing 00:03:34.560 --> 00:03:39.920 and that those savings will be used for 00:03:37.719 --> 00:03:41.800 investment. Okay? 00:03:39.919 --> 00:03:43.599 But investment is nothing else than 00:03:41.800 --> 00:03:44.920 capital accumulation. 00:03:43.599 --> 00:03:47.079 So 00:03:44.919 --> 00:03:49.079 this income will lead to saving which 00:03:47.080 --> 00:03:50.719 will fund investment which will change 00:03:49.080 --> 00:03:52.560 the stock of capital will feed into 00:03:50.719 --> 00:03:54.599 capital stock that will feed into income 00:03:52.560 --> 00:03:56.520 and so on. All this is happening very 00:03:54.599 --> 00:03:59.079 slowly because the capital stock 00:03:56.520 --> 00:04:01.560 accumulates slowly. I mean 00:03:59.080 --> 00:04:03.040 but but but this is what is happening 00:04:01.560 --> 00:04:04.599 and so all the models we're going to 00:04:03.039 --> 00:04:06.799 look at certainly the model we're going 00:04:04.599 --> 00:04:10.400 to look at in this lecture is all about 00:04:06.800 --> 00:04:10.400 this mechanism. Okay? 00:04:10.599 --> 00:04:15.159 So let's remember what we did in the 00:04:12.879 --> 00:04:16.560 previous lecture. We 00:04:15.159 --> 00:04:18.199 uh 00:04:16.560 --> 00:04:19.798 and I'm going to assume that population 00:04:18.199 --> 00:04:20.839 is constant. I'm going to relax that at 00:04:19.798 --> 00:04:22.759 the very end but assume that the 00:04:20.839 --> 00:04:24.159 population is constant and equal to n 00:04:22.759 --> 00:04:26.920 and remember we're not worrying about 00:04:24.160 --> 00:04:31.360 unemployment and stuff like that here. 00:04:26.920 --> 00:04:35.680 Um so output per capita or per person 00:04:31.360 --> 00:04:38.920 uh is y over n and we remember we had an 00:04:35.680 --> 00:04:40.560 production function f of k and n then 00:04:38.920 --> 00:04:43.040 because of constant returns to scale we 00:04:40.560 --> 00:04:46.800 could divide by n on both sides 00:04:43.040 --> 00:04:50.160 everything and we ended up with this 00:04:46.800 --> 00:04:52.199 relationship. So output per person is 00:04:50.160 --> 00:04:54.480 equal to an is is a is an increasing 00:04:52.199 --> 00:04:55.800 function of capital per person. It's an 00:04:54.480 --> 00:04:58.560 increasing function of capital per 00:04:55.800 --> 00:04:59.800 person but it's also concave 00:04:58.560 --> 00:05:01.519 function 00:04:59.800 --> 00:05:02.840 of capital per person. Why is it 00:05:01.519 --> 00:05:04.599 concave? 00:05:02.839 --> 00:05:07.479 That is why is it increasing at a 00:05:04.600 --> 00:05:07.480 smaller pace? 00:05:13.879 --> 00:05:16.319 Uh yeah. 00:05:17.279 --> 00:05:21.759 Decreasing 00:05:18.839 --> 00:05:23.239 marginal product of capital exactly. 00:05:21.759 --> 00:05:25.560 You know, for fixed amount of labor the 00:05:23.240 --> 00:05:27.040 more capital you put in in into 00:05:25.560 --> 00:05:29.120 production well 00:05:27.040 --> 00:05:31.280 output keeps expanding but by less and 00:05:29.120 --> 00:05:34.160 less because it has less and less labor 00:05:31.279 --> 00:05:37.559 to work with each each unit of capital. 00:05:34.160 --> 00:05:38.960 Perfect. That's very important. Uh then 00:05:37.560 --> 00:05:40.560 let's we're going to work in closed 00:05:38.959 --> 00:05:42.159 economy. I haven't opened it. I'm going 00:05:40.560 --> 00:05:43.120 to do that after 00:05:42.160 --> 00:05:44.320 uh 00:05:43.120 --> 00:05:45.480 quiz two. 00:05:44.319 --> 00:05:47.639 Um 00:05:45.480 --> 00:05:50.800 So and I'm going to assume also no 00:05:47.639 --> 00:05:51.680 public deficits so g equal to t capital 00:05:50.800 --> 00:05:54.840 T. 00:05:51.680 --> 00:05:57.439 And in that case then we know that uh 00:05:54.839 --> 00:05:59.359 private investment private savings equal 00:05:57.439 --> 00:06:01.639 to private investment. Okay? That's 00:05:59.360 --> 00:06:05.840 that's the way we derive the IS curve. 00:06:01.639 --> 00:06:07.240 Um so that's that's that's not new. 00:06:05.839 --> 00:06:08.839 I'm going to modify a little bit what we 00:06:07.240 --> 00:06:11.960 did in the short run. 00:06:08.839 --> 00:06:14.279 Uh um and I'm going to assume that that 00:06:11.959 --> 00:06:17.719 savings is proportional to income. So 00:06:14.279 --> 00:06:20.719 savings little s times y. 00:06:17.720 --> 00:06:22.280 Notice that that this is 00:06:20.720 --> 00:06:23.760 is different from what we did in the 00:06:22.279 --> 00:06:25.519 short run. In the short run remember we 00:06:23.759 --> 00:06:28.360 had a c0 floating around. We had a 00:06:25.519 --> 00:06:30.479 constant in the consumption function. So 00:06:28.360 --> 00:06:32.280 savings which was equal to income minus 00:06:30.480 --> 00:06:33.560 consumption also had a constant floating 00:06:32.279 --> 00:06:35.799 around. 00:06:33.560 --> 00:06:37.560 Now that that constant was important in 00:06:35.800 --> 00:06:39.400 the short-run model because you were 00:06:37.560 --> 00:06:41.600 approximating for a bunch of things that 00:06:39.399 --> 00:06:44.759 are not related to short-term income. 00:06:41.600 --> 00:06:48.320 Wealth you know, the price of houses, 00:06:44.759 --> 00:06:50.039 stuff like that. We put all that in that 00:06:48.319 --> 00:06:52.319 constant there. 00:06:50.040 --> 00:06:54.800 When you think about the long run though 00:06:52.319 --> 00:06:57.879 uh most of those things that we excluded 00:06:54.800 --> 00:07:00.480 there asset prices, stuff like that tend 00:06:57.879 --> 00:07:03.600 to scale with output as well. So so this 00:07:00.480 --> 00:07:05.200 is this are inconsistent on the surface 00:07:03.600 --> 00:07:07.879 but if you were to fully work out what 00:07:05.199 --> 00:07:09.920 is behind the c0 in in the consumption 00:07:07.879 --> 00:07:11.360 function then 00:07:09.920 --> 00:07:13.199 this is not a bad approximation. They're 00:07:11.360 --> 00:07:15.879 not that inconsistent because you 00:07:13.199 --> 00:07:18.120 endogenize things that over the long run 00:07:15.879 --> 00:07:20.159 scale with income. I mean, you know, 00:07:18.120 --> 00:07:22.120 wealth tends to rise with income and all 00:07:20.160 --> 00:07:23.760 these things tend to move together. At 00:07:22.120 --> 00:07:25.920 not at high frequency, you can have all 00:07:23.759 --> 00:07:29.159 sort of fluctuations but over the long 00:07:25.920 --> 00:07:31.199 run they tend to scale up together. So 00:07:29.160 --> 00:07:34.160 that's going to be our saving function. 00:07:31.199 --> 00:07:35.639 So that means that we know in 00:07:34.160 --> 00:07:37.520 equilibrium 00:07:35.639 --> 00:07:40.279 this is not investment function. We know 00:07:37.519 --> 00:07:43.519 that in equilibrium investment will be 00:07:40.279 --> 00:07:44.799 equal to uh it will be proportional to 00:07:43.519 --> 00:07:46.919 income. 00:07:44.800 --> 00:07:49.520 Okay? So remember what we were going 00:07:46.920 --> 00:07:51.960 through the box. We had at the top of 00:07:49.519 --> 00:07:53.759 the box we had capital that led to 00:07:51.959 --> 00:07:57.039 output. We're doing everything in terms 00:07:53.759 --> 00:07:58.719 per capita. That less led to saving 00:07:57.040 --> 00:08:02.720 and that 00:07:58.720 --> 00:08:05.480 funded investment. Okay? So that's 00:08:02.720 --> 00:08:08.280 that's what we have. So 00:08:05.480 --> 00:08:10.640 this growth model is really about 00:08:08.279 --> 00:08:12.599 uh these three functional forms and then 00:08:10.639 --> 00:08:13.680 a dynamic equation for the stock of 00:08:12.600 --> 00:08:15.520 capital. 00:08:13.680 --> 00:08:18.199 So 00:08:15.519 --> 00:08:19.599 the evolution of the stock of capital 00:08:18.199 --> 00:08:21.199 capital will increase because of 00:08:19.600 --> 00:08:22.800 investment. 00:08:21.199 --> 00:08:24.920 Uh that's what investment is. It's an 00:08:22.800 --> 00:08:28.800 increase in the stock of capital. 00:08:24.920 --> 00:08:30.600 Uh but but it will also decrease 00:08:28.800 --> 00:08:32.399 as a result of depreciation. I mean 00:08:30.600 --> 00:08:36.200 things do break up, you know. 00:08:32.399 --> 00:08:37.559 Uh once in a while. And so 00:08:36.200 --> 00:08:39.640 and different type of capital have 00:08:37.559 --> 00:08:41.559 different depreciation rates. Equipment 00:08:39.639 --> 00:08:44.120 depreciate much faster than structures 00:08:41.559 --> 00:08:45.599 and buildings and so on. But we're going 00:08:44.120 --> 00:08:47.440 to not going to make those distinctions 00:08:45.600 --> 00:08:50.159 here. But you see this tells you the 00:08:47.440 --> 00:08:52.720 capital stock at t plus one is equal to 00:08:50.159 --> 00:08:54.279 the capital stock we had before minus 00:08:52.720 --> 00:08:57.160 what is depreciated of that stock of 00:08:54.279 --> 00:08:58.279 capital plus any new investment we do 00:08:57.159 --> 00:09:00.439 today. 00:08:58.279 --> 00:09:00.439 Okay? 00:09:00.480 --> 00:09:04.039 In per worker terms and remember that 00:09:02.279 --> 00:09:05.439 for now I'm keeping population growth 00:09:04.039 --> 00:09:07.439 constant 00:09:05.440 --> 00:09:08.840 equal to zero. Not not population growth 00:09:07.440 --> 00:09:12.120 constant. Yeah, constant but equal to 00:09:08.840 --> 00:09:14.879 zero. So population is constant. 00:09:12.120 --> 00:09:18.279 I can divide this both sides by n, you 00:09:14.879 --> 00:09:21.960 know, and I get that capital per worker 00:09:18.279 --> 00:09:24.000 uh per worker or per person 00:09:21.960 --> 00:09:26.280 is equal to this expression here. I did 00:09:24.000 --> 00:09:30.120 two things here. I divided by n and I 00:09:26.279 --> 00:09:33.839 replaced replaced this I function this 00:09:30.120 --> 00:09:35.000 investment for savings. Okay? Because I 00:09:33.840 --> 00:09:37.560 know in equilibrium they have to be 00:09:35.000 --> 00:09:40.080 equal. So I have that. 00:09:37.559 --> 00:09:43.479 Um I can rewrite this, you know, just 00:09:40.080 --> 00:09:45.320 subtract kt over n on both sides and 00:09:43.480 --> 00:09:47.600 then you get at the change in capital 00:09:45.320 --> 00:09:50.680 per person is 00:09:47.600 --> 00:09:53.600 is an increasing function of savings 00:09:50.679 --> 00:09:55.279 and decreasing of depreciation. Okay? So 00:09:53.600 --> 00:09:58.639 the last step 00:09:55.279 --> 00:10:00.039 that is important in this model is to So 00:09:58.639 --> 00:10:01.759 here I have essentially a difference 00:10:00.039 --> 00:10:03.399 equation for capital, but we have an 00:10:01.759 --> 00:10:06.200 output per capita on the right-hand 00:10:03.399 --> 00:10:09.319 side. But it turns out that that I know 00:10:06.200 --> 00:10:11.560 that output per capita per person I said 00:10:09.320 --> 00:10:13.480 per capita per person the same thing 00:10:11.559 --> 00:10:14.919 per worker it's the same thing in this 00:10:13.480 --> 00:10:16.759 part of the course. 00:10:14.919 --> 00:10:19.439 Uh so this 00:10:16.759 --> 00:10:20.879 is equal to uh 00:10:19.440 --> 00:10:23.280 is a function is an increasing and 00:10:20.879 --> 00:10:25.519 concave function of capital 00:10:23.279 --> 00:10:27.439 per person. Okay? 00:10:25.519 --> 00:10:29.279 So this is I would say is the sort of 00:10:27.440 --> 00:10:30.440 fundamental equation of the Solow growth 00:10:29.279 --> 00:10:32.439 model. 00:10:30.440 --> 00:10:33.640 It says the change in the stock of 00:10:32.440 --> 00:10:35.080 capital 00:10:33.639 --> 00:10:38.199 increases 00:10:35.080 --> 00:10:38.200 uh with uh 00:10:38.879 --> 00:10:44.159 with investment of course and decreases 00:10:41.600 --> 00:10:46.600 with depreciation. And both of these 00:10:44.159 --> 00:10:48.799 expressions here are increasing 00:10:46.600 --> 00:10:51.840 functions of the 00:10:48.799 --> 00:10:53.599 stock of capital per person. Okay? 00:10:51.840 --> 00:10:54.759 So let's let's try to understand what is 00:10:53.600 --> 00:10:57.200 in here. 00:10:54.759 --> 00:10:57.200 So 00:10:57.279 --> 00:10:59.838 why 00:10:58.840 --> 00:11:01.839 uh 00:10:59.839 --> 00:11:04.120 so this is linear obviously because 00:11:01.839 --> 00:11:06.520 depreciation is linear. You you say say 00:11:04.120 --> 00:11:09.279 you you lose 5% of your stock of capital 00:11:06.519 --> 00:11:11.519 every year because it breaks down. 00:11:09.279 --> 00:11:13.799 Obviously, the more capital per person 00:11:11.519 --> 00:11:15.240 you have, the more units of capital 00:11:13.799 --> 00:11:17.399 you're going to lose. This is in units 00:11:15.240 --> 00:11:18.960 of capital per person. If you have a 00:11:17.399 --> 00:11:21.639 larger stock of capital, you're going to 00:11:18.960 --> 00:11:23.200 lose 5% of of a larger number is a 00:11:21.639 --> 00:11:25.039 larger number. So this and this is 00:11:23.200 --> 00:11:27.160 proportional is linear. 00:11:25.039 --> 00:11:30.559 Now this one remember this comes here 00:11:27.159 --> 00:11:33.240 from the saving function and this term 00:11:30.559 --> 00:11:33.919 here is equal to income per person. 00:11:33.240 --> 00:11:36.200 Uh 00:11:33.919 --> 00:11:38.039 now suppose that that you start in a 00:11:36.200 --> 00:11:40.680 situation where the capital stock is 00:11:38.039 --> 00:11:42.439 relatively low and this 00:11:40.679 --> 00:11:44.399 is positive. 00:11:42.440 --> 00:11:45.760 What does it mean that this is positive? 00:11:44.399 --> 00:11:47.879 I mean the implication of this being 00:11:45.759 --> 00:11:50.519 positive is that the stock stock of 00:11:47.879 --> 00:11:51.879 capital per person will be growing. 00:11:50.519 --> 00:11:54.960 But what does it mean that this is 00:11:51.879 --> 00:11:54.960 positive in words? 00:11:56.159 --> 00:12:00.079 I mean if you have a stock of capital, 00:11:58.519 --> 00:12:01.480 there are things that reduce the stock 00:12:00.080 --> 00:12:03.520 of capital and there are things that 00:12:01.480 --> 00:12:05.360 increase the stock of capital. 00:12:03.519 --> 00:12:06.759 This is the thing that increases the 00:12:05.360 --> 00:12:08.879 stock of capital that's the thing that 00:12:06.759 --> 00:12:10.960 reduces the stock of capital. 00:12:08.879 --> 00:12:13.200 So 00:12:10.960 --> 00:12:15.720 if this 00:12:13.200 --> 00:12:17.759 is greater than that what is that What 00:12:15.720 --> 00:12:19.680 does that mean? It's that means this is 00:12:17.759 --> 00:12:22.360 positive, but in words 00:12:19.679 --> 00:12:22.359 what is happening? 00:12:31.159 --> 00:12:37.000 Let me simplify it. This is remember 00:12:33.080 --> 00:12:37.000 this is just investment per person. 00:12:39.120 --> 00:12:42.159 Well, this just says 00:12:41.000 --> 00:12:44.279 that 00:12:42.159 --> 00:12:46.799 this economy in this economy there is 00:12:44.279 --> 00:12:49.240 more investment than destruction of 00:12:46.799 --> 00:12:51.439 capital due to depreciation. 00:12:49.240 --> 00:12:53.279 Okay? That's what this means. 00:12:51.440 --> 00:12:55.120 This is investment 00:12:53.279 --> 00:12:57.319 and and this is positive means that the 00:12:55.120 --> 00:12:59.039 investment that which is a function of 00:12:57.320 --> 00:13:01.520 saving the saving rate and stuff like 00:12:59.039 --> 00:13:03.439 that uh is a function of the funding 00:13:01.519 --> 00:13:04.960 available for investment 00:13:03.440 --> 00:13:07.120 is equal to the funding available for 00:13:04.960 --> 00:13:07.800 investment. Uh 00:13:07.120 --> 00:13:09.600 uh 00:13:07.799 --> 00:13:12.240 if this is positive, well this is 00:13:09.600 --> 00:13:14.120 greater than the stock of capital. 00:13:12.240 --> 00:13:15.879 Another way of saying it you need a 00:13:14.120 --> 00:13:17.440 minimum level of investment in an 00:13:15.879 --> 00:13:20.039 economy to maintain the stock of 00:13:17.440 --> 00:13:20.040 capital. 00:13:20.320 --> 00:13:24.040 The minimum level of investment that you 00:13:21.679 --> 00:13:26.679 need to maintain the stock of capital is 00:13:24.039 --> 00:13:27.838 equal to the depreciation. So 10 machine 00:13:26.679 --> 00:13:30.199 breaks 00:13:27.839 --> 00:13:31.520 you need to invest at least 10 machines 00:13:30.200 --> 00:13:34.320 in order to maintain the stock of 00:13:31.519 --> 00:13:36.000 capital constant. Okay? 00:13:34.320 --> 00:13:37.600 Now if this is positive, it means you're 00:13:36.000 --> 00:13:39.240 investing more than the machines that 00:13:37.600 --> 00:13:41.000 are breaking down. 00:13:39.240 --> 00:13:42.519 Now suppose you start in a situation 00:13:41.000 --> 00:13:44.159 where that's the case. 00:13:42.519 --> 00:13:45.519 So that means the stock of capital is 00:13:44.159 --> 00:13:48.120 growing. 00:13:45.519 --> 00:13:49.919 I suppose I ask you the next period do 00:13:48.120 --> 00:13:51.000 you think that gap will be larger or 00:13:49.919 --> 00:13:53.879 smaller 00:13:51.000 --> 00:13:53.879 than it used to be? 00:14:01.879 --> 00:14:04.399 Yeah, actually that's not a great 00:14:03.360 --> 00:14:05.560 question. 00:14:04.399 --> 00:14:07.759 Well 00:14:05.559 --> 00:14:10.519 because I'm not doing it in the right 00:14:07.759 --> 00:14:12.159 units for that. 00:14:10.519 --> 00:14:14.879 Let me ask you a 00:14:12.159 --> 00:14:16.838 variation of that question. Suppose we 00:14:14.879 --> 00:14:20.039 keep going. 00:14:16.839 --> 00:14:23.280 After a while, do you think that number 00:14:20.039 --> 00:14:25.159 will get larger or smaller? After a 00:14:23.279 --> 00:14:26.559 after let it run for a little for for 00:14:25.159 --> 00:14:28.679 quite a while. 00:14:26.559 --> 00:14:30.119 Do you think that number will So 00:14:28.679 --> 00:14:32.799 remember I said we start with some stock 00:14:30.120 --> 00:14:33.919 of capital. This is positive. 00:14:32.799 --> 00:14:35.799 If this is positive, it means that the 00:14:33.919 --> 00:14:37.759 capital stock is growing. That means 00:14:35.799 --> 00:14:41.039 this guy is growing and that guy is 00:14:37.759 --> 00:14:42.720 growing. And they're growing equally. 00:14:41.039 --> 00:14:45.719 But after a while, do you think this 00:14:42.720 --> 00:14:47.720 number will get smaller or bigger? 00:14:45.720 --> 00:14:51.639 After a long while just to make sure 00:14:47.720 --> 00:14:51.639 that my approximation is not bad here. 00:14:57.720 --> 00:15:00.360 Exactly. It's going to get smaller 00:14:59.320 --> 00:15:03.079 because 00:15:00.360 --> 00:15:04.639 this guy keeps growing linearly 00:15:03.078 --> 00:15:06.879 with the stock of capital and this one 00:15:04.639 --> 00:15:08.879 is not. It's concave, you know? 00:15:06.879 --> 00:15:10.439 At some point this income sort of you 00:15:08.879 --> 00:15:12.559 need to put a lot of capital for for 00:15:10.440 --> 00:15:14.400 income to keep rising and therefore for 00:15:12.559 --> 00:15:16.439 saving to keep rising and therefore for 00:15:14.399 --> 00:15:19.439 investment to keep rising. And at some 00:15:16.440 --> 00:15:20.160 point yes it won't be able to 00:15:19.440 --> 00:15:21.400 uh 00:15:20.159 --> 00:15:22.919 to really grow. I mean you're going to 00:15:21.399 --> 00:15:24.838 be using all your investment really to 00:15:22.919 --> 00:15:26.639 maintain the stock of capital. 00:15:24.839 --> 00:15:29.959 That's sort of the logic 00:15:26.639 --> 00:15:32.039 of the Solow model. 00:15:29.958 --> 00:15:33.479 And it's all in this diagram. So this is 00:15:32.039 --> 00:15:36.319 diagram you should really really 00:15:33.480 --> 00:15:38.320 understand well and control it and play 00:15:36.320 --> 00:15:42.120 with it and all that. It's the 00:15:38.320 --> 00:15:44.520 equivalent to your IS-LM model in in in 00:15:42.120 --> 00:15:46.759 the first part of the course. 00:15:44.519 --> 00:15:51.000 So look at what you have here. 00:15:46.759 --> 00:15:53.559 So I'm going to plot output per worker 00:15:51.000 --> 00:15:55.958 per worker per person against capital 00:15:53.559 --> 00:15:58.599 per worker here. 00:15:55.958 --> 00:16:01.519 And so 00:15:58.600 --> 00:16:02.920 this red line here 00:16:01.519 --> 00:16:05.679 is just 00:16:02.919 --> 00:16:07.240 the depreciation. Okay? This 00:16:05.679 --> 00:16:10.039 term here. 00:16:07.240 --> 00:16:12.560 And that's is a linear function of the 00:16:10.039 --> 00:16:14.838 capital per worker. Okay? That's what it 00:16:12.559 --> 00:16:14.838 is. 00:16:15.078 --> 00:16:18.000 Uh 00:16:15.919 --> 00:16:20.039 the blue line here 00:16:18.000 --> 00:16:22.679 is output per worker 00:16:20.039 --> 00:16:24.879 which as we said is a concave function 00:16:22.679 --> 00:16:26.759 of K over N. Remember I showed you that 00:16:24.879 --> 00:16:28.120 production function last in the last in 00:16:26.759 --> 00:16:29.319 the previous lecture. 00:16:28.120 --> 00:16:30.320 There you are. 00:16:29.320 --> 00:16:32.280 Okay? 00:16:30.320 --> 00:16:34.520 What is the green line? 00:16:32.279 --> 00:16:36.600 Is investment per worker which is equal 00:16:34.519 --> 00:16:40.679 to saving per worker and saving per 00:16:36.600 --> 00:16:43.279 worker is little s the saving rate times 00:16:40.679 --> 00:16:46.639 uh output. So it's little s which is a 00:16:43.279 --> 00:16:49.199 number like 0.1 if if if we're talking 00:16:46.639 --> 00:16:51.279 about the US and you know 0.4 if we're 00:16:49.200 --> 00:16:54.680 talking about Singapore it varies a lot 00:16:51.279 --> 00:16:57.480 across countries. But but uh 00:16:54.679 --> 00:16:59.239 but so this this green line here is 00:16:57.480 --> 00:17:00.839 nothing else than this blue line 00:16:59.240 --> 00:17:05.200 multiplied by a number that is less than 00:17:00.839 --> 00:17:05.200 one. That's the reason it's lower. Okay? 00:17:06.039 --> 00:17:10.759 Okay, good. So the point I was 00:17:08.119 --> 00:17:12.078 describing before is was a point like 00:17:10.759 --> 00:17:13.078 this. 00:17:12.078 --> 00:17:15.838 Remember? 00:17:13.078 --> 00:17:17.759 Uh the point that I was describing 00:17:15.838 --> 00:17:20.559 suppose the economy starts in a point 00:17:17.759 --> 00:17:21.920 like this one K0 over N. 00:17:20.559 --> 00:17:23.480 Well 00:17:21.920 --> 00:17:25.519 and I want to understand the dynamics of 00:17:23.480 --> 00:17:26.599 this economy. How will it grow over 00:17:25.519 --> 00:17:27.639 time? 00:17:26.599 --> 00:17:31.199 So 00:17:27.640 --> 00:17:32.360 what you have see here is that that 00:17:31.200 --> 00:17:36.200 uh 00:17:32.359 --> 00:17:38.639 at this level of capital per worker 00:17:36.200 --> 00:17:39.360 investment is greater than 00:17:38.640 --> 00:17:41.480 uh 00:17:39.359 --> 00:17:43.839 than depreciation. 00:17:41.480 --> 00:17:45.360 So that's exactly a situation where this 00:17:43.839 --> 00:17:46.919 is positive. 00:17:45.359 --> 00:17:49.959 Okay? 00:17:46.920 --> 00:17:52.679 That distance here 00:17:49.960 --> 00:17:52.679 is that. 00:17:52.839 --> 00:17:55.559 Okay? 00:17:54.000 --> 00:17:57.200 And the reason I sort of 00:17:55.559 --> 00:17:58.639 say I'm not going to do any local 00:17:57.200 --> 00:18:01.360 analysis because we could have a started 00:17:58.640 --> 00:18:03.560 with a K over zero over here and then 00:18:01.359 --> 00:18:05.039 that number is growing, but it's growing 00:18:03.559 --> 00:18:07.200 if you were to normalize by the stock of 00:18:05.039 --> 00:18:08.759 capital is is is declining. That's 00:18:07.200 --> 00:18:11.679 that's that but I didn't want to do that 00:18:08.759 --> 00:18:12.679 then. But now that's what I So let's 00:18:11.679 --> 00:18:14.720 look at this case. You're you're in a 00:18:12.679 --> 00:18:16.400 situation where this is positive. If 00:18:14.720 --> 00:18:18.839 this is positive 00:18:16.400 --> 00:18:21.600 it means the capital stock per worker is 00:18:18.839 --> 00:18:23.199 growing. So you're moving to the right. 00:18:21.599 --> 00:18:24.199 In the next period you're going to be 00:18:23.200 --> 00:18:25.319 here. 00:18:24.200 --> 00:18:26.519 That 00:18:25.319 --> 00:18:27.678 that means the capital stock keeps 00:18:26.519 --> 00:18:30.918 growing 00:18:27.679 --> 00:18:33.000 but by a smaller steps. 00:18:30.919 --> 00:18:33.679 Eventually 00:18:33.000 --> 00:18:36.119 uh 00:18:33.679 --> 00:18:39.040 the investment is entirely used 00:18:36.119 --> 00:18:39.039 for uh 00:18:39.079 --> 00:18:42.759 recovering from the depreciation of 00:18:40.559 --> 00:18:45.240 capital. So covering the depreciation of 00:18:42.759 --> 00:18:48.359 capital. And that point the capital 00:18:45.240 --> 00:18:52.519 stock stock stops growing. We call that 00:18:48.359 --> 00:18:54.639 a steady state stationary state. We stop 00:18:52.519 --> 00:18:55.679 Okay? So that's the steady state of this 00:18:54.640 --> 00:18:56.960 model. 00:18:55.679 --> 00:18:59.320 That means 00:18:56.960 --> 00:19:00.960 this economy regardless of where is I do 00:18:59.319 --> 00:19:02.399 analysis from the other side. Suppose 00:19:00.960 --> 00:19:04.480 you start from a situation like this. 00:19:02.400 --> 00:19:06.600 You start with a lot of capital. 00:19:04.480 --> 00:19:08.880 Okay? Well, if you start with a lot of 00:19:06.599 --> 00:19:10.399 capital in this economy 00:19:08.880 --> 00:19:12.159 what happens 00:19:10.400 --> 00:19:13.880 when here? 00:19:12.159 --> 00:19:15.800 Well, what happens here is that the 00:19:13.880 --> 00:19:17.360 investment you're putting to the ground 00:19:15.799 --> 00:19:19.960 in this economy is less than what you 00:19:17.359 --> 00:19:21.799 need to maintain the stock of capital 00:19:19.960 --> 00:19:23.319 which is depreciation. 00:19:21.799 --> 00:19:25.480 And that means the stock of capital will 00:19:23.319 --> 00:19:27.559 be shrinking over time. 00:19:25.480 --> 00:19:30.240 Okay? You're moving that way. 00:19:27.559 --> 00:19:33.359 So regardless of where you start in this 00:19:30.240 --> 00:19:35.880 economy if I I ask you the question 100 00:19:33.359 --> 00:19:38.119 years from now, where are you? 00:19:35.880 --> 00:19:39.560 You I tell you tell me I don't need to 00:19:38.119 --> 00:19:41.359 know where you start from. I know that 00:19:39.559 --> 00:19:43.319 we're going to end up around there. 00:19:41.359 --> 00:19:44.479 You can either you start from here, you 00:19:43.319 --> 00:19:46.200 go there 00:19:44.480 --> 00:19:47.960 from here you go there and so on. That's 00:19:46.200 --> 00:19:49.880 the reason we call this a steady state. 00:19:47.960 --> 00:19:52.759 This is where you converge in the long 00:19:49.880 --> 00:19:52.760 run. Okay? 00:19:55.000 --> 00:19:57.920 Now, 00:19:56.319 --> 00:19:59.319 this is already interesting because it 00:19:57.920 --> 00:20:01.400 tells you 00:19:59.319 --> 00:20:04.079 you know, at this mo- at this point 00:20:01.400 --> 00:20:06.000 here, the economy was growing. You know, 00:20:04.079 --> 00:20:08.599 the capital stock was growing and and 00:20:06.000 --> 00:20:10.400 and and the and output was growing. You 00:20:08.599 --> 00:20:11.480 see, the capital is if you start from 00:20:10.400 --> 00:20:13.120 here, 00:20:11.480 --> 00:20:15.000 the capital stock is growing, well, 00:20:13.119 --> 00:20:17.759 output is also growing. 00:20:15.000 --> 00:20:20.039 Okay? You're moving up there. 00:20:17.759 --> 00:20:22.359 Okay? So, you had growth. 00:20:20.039 --> 00:20:23.680 That kind of growth we call transitional 00:20:22.359 --> 00:20:26.879 growth. 00:20:23.680 --> 00:20:28.480 You know, it goes from one point to 00:20:26.880 --> 00:20:30.680 another point. It's not a permanent 00:20:28.480 --> 00:20:32.279 growth. It's transitional growth. 00:20:30.680 --> 00:20:34.560 It's the fact that you were away from 00:20:32.279 --> 00:20:37.399 your steady state and then you're going 00:20:34.559 --> 00:20:39.119 converging towards your steady state. 00:20:37.400 --> 00:20:40.720 A lot of the growth we observe and the 00:20:39.119 --> 00:20:41.919 difference of growth we observe across 00:20:40.720 --> 00:20:44.519 countries, remember I showed you the 00:20:41.920 --> 00:20:47.600 downward sloping curves and all that, 00:20:44.519 --> 00:20:49.559 is as a result of that. Poorer economies 00:20:47.599 --> 00:20:50.279 tend to have lower capital 00:20:49.559 --> 00:20:53.399 uh 00:20:50.279 --> 00:20:55.559 capital labor capital employment ratios, 00:20:53.400 --> 00:20:57.600 capital population ratios, and therefore 00:20:55.559 --> 00:20:58.799 they they tend to grow faster because 00:20:57.599 --> 00:21:00.159 they're catching up with their steady 00:20:58.799 --> 00:21:01.879 state. 00:21:00.160 --> 00:21:03.120 Very advanced economies that have been 00:21:01.880 --> 00:21:05.720 more or less in the same place for a 00:21:03.119 --> 00:21:08.479 long time are moving around there. 00:21:05.720 --> 00:21:10.160 So, there's less catching up growth. 00:21:08.480 --> 00:21:12.480 And that's the main responsible for the 00:21:10.160 --> 00:21:14.600 the downward sloping curve I showed you 00:21:12.480 --> 00:21:16.200 within OECD countries and even broader 00:21:14.599 --> 00:21:18.719 than that. Africa was a little of a 00:21:16.200 --> 00:21:18.720 problem there. 00:21:18.880 --> 00:21:23.360 Okay. 00:21:20.640 --> 00:21:24.360 So, that's This is an important model. 00:21:23.359 --> 00:21:26.000 Okay for you. 00:21:24.359 --> 00:21:28.399 Important diagram. 00:21:26.000 --> 00:21:30.319 Let's let's play a little with it. So, 00:21:28.400 --> 00:21:32.560 suppose that, you know, at the time, 00:21:30.319 --> 00:21:34.000 this is a very simple model, but 00:21:32.559 --> 00:21:36.599 at the time, 00:21:34.000 --> 00:21:37.680 the the view was that uh 00:21:36.599 --> 00:21:40.319 well, 00:21:37.680 --> 00:21:42.519 what really supports growth is saving. 00:21:40.319 --> 00:21:44.759 So, economies that save a lot 00:21:42.519 --> 00:21:46.759 grow a lot. And this sort of sort of 00:21:44.759 --> 00:21:48.720 makes sense here because 00:21:46.759 --> 00:21:50.440 investment, which is what leads to 00:21:48.720 --> 00:21:52.880 capital accumulation, is entirely funded 00:21:50.440 --> 00:21:54.200 by savings. It makes sense. 00:21:52.880 --> 00:21:55.920 You have more saving, you should grow 00:21:54.200 --> 00:21:58.240 more. 00:21:55.920 --> 00:22:00.640 Okay, so let's This is something we can 00:21:58.240 --> 00:22:03.400 do an experiment. Suppose you start at 00:22:00.640 --> 00:22:06.720 at at a steady state, if you will. 00:22:03.400 --> 00:22:09.000 And now we increase the saving rate. 00:22:06.720 --> 00:22:10.600 What moves? 00:22:09.000 --> 00:22:12.319 Which curve This is the kind of things 00:22:10.599 --> 00:22:13.919 you should know when you work with this 00:22:12.319 --> 00:22:16.200 model. 00:22:13.920 --> 00:22:17.720 If I change the saving rate, which curve 00:22:16.200 --> 00:22:19.200 moves 00:22:17.720 --> 00:22:20.960 in this model? 00:22:19.200 --> 00:22:24.200 Let me go one by one. 00:22:20.960 --> 00:22:24.200 Does the red line move? 00:22:24.880 --> 00:22:27.679 No, has nothing to do with savings. 00:22:26.559 --> 00:22:29.799 That's to do with depreciation. If I 00:22:27.679 --> 00:22:32.480 move the depreciation rate, that curve 00:22:29.799 --> 00:22:35.159 will move, but not 00:22:32.480 --> 00:22:37.799 Will the production function move? 00:22:35.160 --> 00:22:39.360 No. So, the blue line cannot move. 00:22:37.799 --> 00:22:41.200 All that will move is the green line 00:22:39.359 --> 00:22:43.678 because the green line is the saving 00:22:41.200 --> 00:22:46.000 rate times the 00:22:43.679 --> 00:22:47.600 the the blue line. So, if I increase the 00:22:46.000 --> 00:22:49.079 saving rate, I'm going to move the green 00:22:47.599 --> 00:22:52.439 line up. 00:22:49.079 --> 00:22:52.439 Okay? And that's what we have here. 00:22:52.480 --> 00:22:57.079 So, you see what happens is you start 00:22:54.839 --> 00:22:59.319 for for This was a steady state for this 00:22:57.079 --> 00:23:00.599 saving rate in this economy. 00:22:59.319 --> 00:23:02.960 Now, all of the sudden this economy 00:23:00.599 --> 00:23:05.159 starts saving more. 00:23:02.960 --> 00:23:06.840 What happens then? 00:23:05.160 --> 00:23:09.200 This tells you very much the story of 00:23:06.839 --> 00:23:11.599 Asia, the Asian miracle 00:23:09.200 --> 00:23:14.400 of the '60s, '70s, and so on is very 00:23:11.599 --> 00:23:16.240 much something like that. 00:23:14.400 --> 00:23:18.360 A little more complicated, but, you 00:23:16.240 --> 00:23:18.359 know, 00:23:18.480 --> 00:23:22.640 a big part of what explains sort of the 00:23:20.559 --> 00:23:23.879 fast growth of Asia 00:23:22.640 --> 00:23:25.960 uh 00:23:23.880 --> 00:23:28.000 during that period 00:23:25.960 --> 00:23:29.640 uh is that something like that happened. 00:23:28.000 --> 00:23:30.880 Now, why the saving rate increases and 00:23:29.640 --> 00:23:32.840 so on, that's all very interesting and 00:23:30.880 --> 00:23:34.600 so on, but but it's not what I want to 00:23:32.839 --> 00:23:35.879 discuss today. 00:23:34.599 --> 00:23:37.359 So, 00:23:35.880 --> 00:23:38.679 but what happens here then? So, what 00:23:37.359 --> 00:23:40.439 happens See, this economy was in a 00:23:38.679 --> 00:23:42.320 steady state, so there was no growth. It 00:23:40.440 --> 00:23:43.519 was growing at zero in steady state, you 00:23:42.319 --> 00:23:44.879 know? 00:23:43.519 --> 00:23:47.960 Because 00:23:44.880 --> 00:23:49.960 this says in a steady state output per 00:23:47.960 --> 00:23:51.600 per worker per remains constant and 00:23:49.960 --> 00:23:52.960 since we have no population work and 00:23:51.599 --> 00:23:54.319 growth, then that means output is not 00:23:52.960 --> 00:23:55.759 growing either. 00:23:54.319 --> 00:23:58.159 Okay? The only way you can have that 00:23:55.759 --> 00:23:59.960 ratio constant with the denominator not 00:23:58.160 --> 00:24:02.279 moving is that the numerator is not 00:23:59.960 --> 00:24:03.759 moving either. Okay? 00:24:02.279 --> 00:24:05.759 Okay, good. 00:24:03.759 --> 00:24:07.359 So, now 00:24:05.759 --> 00:24:11.000 boom, all of the sudden we get a higher 00:24:07.359 --> 00:24:13.759 saving rate. So, what happens now? 00:24:11.000 --> 00:24:13.759 What reacts? 00:24:14.319 --> 00:24:18.000 So, the saving rates go up. It's a 00:24:16.079 --> 00:24:19.439 closed economy, it means the investment 00:24:18.000 --> 00:24:21.079 rate will go up. 00:24:19.440 --> 00:24:24.120 Okay? 00:24:21.079 --> 00:24:24.119 What happens now? 00:24:29.640 --> 00:24:33.600 What does that gap tell you? 00:24:35.000 --> 00:24:39.319 Now, you have a positive gap there, 00:24:37.319 --> 00:24:41.319 which means you're investing more than 00:24:39.319 --> 00:24:42.919 the the what you need in order to 00:24:41.319 --> 00:24:45.039 maintain the stock of capital at the 00:24:42.920 --> 00:24:46.720 previous steady state. 00:24:45.039 --> 00:24:48.319 So, that means the stock of capital is 00:24:46.720 --> 00:24:49.799 going to start growing to the right. 00:24:48.319 --> 00:24:50.639 It's going to start growing. 00:24:49.799 --> 00:24:53.359 Okay? 00:24:50.640 --> 00:24:56.480 And as the stock of capital grows, then 00:24:53.359 --> 00:24:58.439 output per capita also grows. 00:24:56.480 --> 00:25:01.960 And this will keep happening until you 00:24:58.440 --> 00:25:01.960 reach the new steady state. 00:25:02.279 --> 00:25:07.079 So, a higher saving rate, so important 00:25:05.279 --> 00:25:08.519 conclusion there. This This as simple as 00:25:07.079 --> 00:25:10.919 it is 00:25:08.519 --> 00:25:11.720 proves something. 00:25:10.920 --> 00:25:13.800 Uh 00:25:11.720 --> 00:25:15.279 that, you know, the conventional wisdom 00:25:13.799 --> 00:25:18.319 that a higher saving rate would give you 00:25:15.279 --> 00:25:19.720 sustained growth, higher growth, 00:25:18.319 --> 00:25:21.559 isn't really true. 00:25:19.720 --> 00:25:23.079 And not certainly not in this model. 00:25:21.559 --> 00:25:24.480 Eventually, you'll go back to growth 00:25:23.079 --> 00:25:26.960 equal to zero. 00:25:24.480 --> 00:25:29.240 Okay? When you reach a new steady state, 00:25:26.960 --> 00:25:31.279 you're going to be also growing at zero. 00:25:29.240 --> 00:25:33.039 Okay? 00:25:31.279 --> 00:25:34.559 What is true, though, 00:25:33.039 --> 00:25:38.480 is that you get what again what is 00:25:34.559 --> 00:25:39.759 called transitional growth. It goes Oh, 00:25:38.480 --> 00:25:42.480 here you're going to start growing very 00:25:39.759 --> 00:25:44.240 fast, in fact. Okay? And then you're 00:25:42.480 --> 00:25:45.880 going to keep growing at a low slow 00:25:44.240 --> 00:25:47.120 lower pace until you go back to zero, 00:25:45.880 --> 00:25:48.720 but you're going to get lots of growth 00:25:47.119 --> 00:25:50.799 in the transition 00:25:48.720 --> 00:25:52.799 as a result of that. And it turns out in 00:25:50.799 --> 00:25:56.639 the data when you're looking at 20 30 00:25:52.799 --> 00:25:58.519 years of data, it's difficult to uh 00:25:56.640 --> 00:26:00.240 disentangle sort of very permanent rates 00:25:58.519 --> 00:26:01.440 of growth versus transitional rate of 00:26:00.240 --> 00:26:02.519 growth. 00:26:01.440 --> 00:26:04.080 This is one of the things that has 00:26:02.519 --> 00:26:05.759 concerned China quite a bit, you know, 00:26:04.079 --> 00:26:07.039 they have been they grow very very fast. 00:26:05.759 --> 00:26:09.079 They have been growing very very fast 00:26:07.039 --> 00:26:10.920 for a long time, but it's very clear 00:26:09.079 --> 00:26:12.639 it's becoming harder and harder for them 00:26:10.920 --> 00:26:14.200 to grow at the type of rate of growth 00:26:12.640 --> 00:26:15.600 that they had in the 00:26:14.200 --> 00:26:17.880 20 years ago. 00:26:15.599 --> 00:26:18.839 Okay? They had rates of growth of 15% or 00:26:17.880 --> 00:26:21.000 so. 00:26:18.839 --> 00:26:22.480 They had very high They had a very low 00:26:21.000 --> 00:26:24.240 initial capital 00:26:22.480 --> 00:26:26.880 population ratio, 00:26:24.240 --> 00:26:29.400 big population, little capital, and 00:26:26.880 --> 00:26:32.160 enormous saving rates. 00:26:29.400 --> 00:26:33.880 So, so they grew very very fast. 00:26:32.160 --> 00:26:36.560 They had like the green line very close 00:26:33.880 --> 00:26:39.679 to to the blue line, the capital stock 00:26:36.559 --> 00:26:40.879 very low, so they grew very very fast. 00:26:39.679 --> 00:26:42.320 But they have been growing very fast for 00:26:40.880 --> 00:26:43.720 a very long period of time, so now it's 00:26:42.319 --> 00:26:45.119 getting a lot harder because they're 00:26:43.720 --> 00:26:47.920 getting closer and closer to their 00:26:45.119 --> 00:26:49.439 steady state. That's the issue. Okay. 00:26:47.920 --> 00:26:50.440 There are other sources of growth, and 00:26:49.440 --> 00:26:52.000 that's what we're going to talk about in 00:26:50.440 --> 00:26:54.440 the next lecture, 00:26:52.000 --> 00:26:56.799 but but this This is something called 00:26:54.440 --> 00:27:00.240 the easy part of growth. 00:26:56.799 --> 00:27:00.240 It's sort of running out in China. 00:27:03.920 --> 00:27:06.160 Okay. 00:27:09.720 --> 00:27:13.120 And it has to run out 00:27:11.400 --> 00:27:15.440 in all developed economies for quite a 00:27:13.119 --> 00:27:15.439 while. 00:27:17.119 --> 00:27:21.399 Um 00:27:18.920 --> 00:27:21.400 good. 00:27:22.079 --> 00:27:25.399 Is this clear? 00:27:23.799 --> 00:27:28.200 It's important. I mean, a question like 00:27:25.400 --> 00:27:30.120 that is guaranteed in your quiz. 00:27:28.200 --> 00:27:31.519 It's 81. 00:27:30.119 --> 00:27:33.479 What happens if the saving rate does 00:27:31.519 --> 00:27:34.679 something? 00:27:33.480 --> 00:27:38.759 So, 00:27:34.679 --> 00:27:40.320 so so this is a plot over time or um 00:27:38.759 --> 00:27:42.200 so, this is a case in which you were in 00:27:40.319 --> 00:27:44.559 a steady state and at time T the saving 00:27:42.200 --> 00:27:47.840 rate goes up. 00:27:44.559 --> 00:27:50.720 S1 greater than S0 jump. 00:27:47.839 --> 00:27:52.359 Then output cannot jump. 00:27:50.720 --> 00:27:54.679 So, the saving rate goes up, but output 00:27:52.359 --> 00:27:55.919 can- cannot jump at day zero. Why? 00:27:54.679 --> 00:27:59.400 Why is it that output doesn't jump 00:27:55.920 --> 00:27:59.400 immediately to a new steady state? 00:28:02.079 --> 00:28:04.240 You know, 00:28:02.920 --> 00:28:05.880 this is the 00:28:04.240 --> 00:28:07.200 I'm I'm saying 00:28:05.880 --> 00:28:08.800 this is what will happen to output. 00:28:07.200 --> 00:28:10.559 You're going to start growing very fast 00:28:08.799 --> 00:28:12.639 early on, and then you keep growing, 00:28:10.559 --> 00:28:13.799 keep growing at a slower and lower pace 00:28:12.640 --> 00:28:15.120 because of decreasing returns to 00:28:13.799 --> 00:28:16.799 capital, 00:28:15.119 --> 00:28:19.079 uh and eventually you'll converge to a 00:28:16.799 --> 00:28:21.639 new steady state with with a rate of 00:28:19.079 --> 00:28:25.199 growth equal to zero, well, like the one 00:28:21.640 --> 00:28:26.840 you had before this savings shock. 00:28:25.200 --> 00:28:28.519 And the question I'm asking now is, why 00:28:26.839 --> 00:28:32.359 doesn't out- Why does output have to do 00:28:28.519 --> 00:28:32.359 this? Why Why doesn't it just jump? 00:28:35.319 --> 00:28:40.039 What would What is the only variable 00:28:36.839 --> 00:28:40.039 that could make it jump? 00:28:41.440 --> 00:28:44.000 Well, you need to look at the production 00:28:42.599 --> 00:28:45.158 function. 00:28:44.000 --> 00:28:47.200 The production function is a function of 00:28:45.159 --> 00:28:48.760 K over N. N is fixed. The only thing 00:28:47.200 --> 00:28:50.039 that can make it jump is if the capital 00:28:48.759 --> 00:28:52.079 stock jumps. 00:28:50.039 --> 00:28:53.559 But the capital stock's not jumping. 00:28:52.079 --> 00:28:54.918 That's a stock. 00:28:53.559 --> 00:28:56.720 And in order to accumulate a larger 00:28:54.919 --> 00:28:58.759 stock of the new steady state, you're 00:28:56.720 --> 00:29:00.679 going to go through a lot of flows. 00:28:58.759 --> 00:29:01.879 That's investment. You know, every year 00:29:00.679 --> 00:29:03.200 you're going to be adding a little more 00:29:01.880 --> 00:29:03.840 to the stock of capital on net or or or 00:29:03.200 --> 00:29:05.120 or 00:29:03.839 --> 00:29:06.639 That's the way you grow. You It's not 00:29:05.119 --> 00:29:10.119 that all of the sudden 00:29:06.640 --> 00:29:12.480 your stock of capital jumps. 00:29:10.119 --> 00:29:14.199 That's very much because this is a 00:29:12.480 --> 00:29:15.599 closed economy. If you're in an open 00:29:14.200 --> 00:29:18.240 economy, the capital stock can move a 00:29:15.599 --> 00:29:20.119 lot faster in a transition because you 00:29:18.240 --> 00:29:22.519 can borrow from abroad. You don't need 00:29:20.119 --> 00:29:24.399 to fund it all with domestic 00:29:22.519 --> 00:29:26.079 sources. And in fact, that's what 00:29:24.400 --> 00:29:28.560 typically happens 00:29:26.079 --> 00:29:31.079 in in in emerging markets and so on is 00:29:28.559 --> 00:29:32.639 they typically borrow for a long time. 00:29:31.079 --> 00:29:33.799 Problem is that they tend to consume it 00:29:32.640 --> 00:29:35.480 rather than invest it, and that's the 00:29:33.799 --> 00:29:37.918 reason you end up in financial crisis 00:29:35.480 --> 00:29:39.480 and so on. But but but in principle, 00:29:37.919 --> 00:29:41.560 things could go much faster if you have 00:29:39.480 --> 00:29:43.319 an open economy and and you have capital 00:29:41.559 --> 00:29:46.639 inflows into your country. But that 00:29:43.319 --> 00:29:48.879 you'll we'll talk about more about that 00:29:46.640 --> 00:29:50.240 five or six six lectures from Anyways, 00:29:48.880 --> 00:29:52.400 but this is what happens when I'm 00:29:50.240 --> 00:29:54.480 increasing the saving rate. So, yes, it 00:29:52.400 --> 00:29:56.440 affects the rate of growth of the 00:29:54.480 --> 00:29:58.319 economy during the transition, 00:29:56.440 --> 00:30:00.759 uh but but not in the long run. Now, 00:29:58.319 --> 00:30:02.519 this transition can be very long. 00:30:00.759 --> 00:30:06.160 Okay? 00:30:02.519 --> 00:30:07.119 Now, what about consumption? So, so 00:30:06.160 --> 00:30:09.200 uh uh 00:30:07.119 --> 00:30:11.119 invariably, and there's no way around 00:30:09.200 --> 00:30:11.680 that, if if 00:30:11.119 --> 00:30:14.279 uh 00:30:11.680 --> 00:30:16.640 given a technology and so on, if the 00:30:14.279 --> 00:30:18.279 saving rate goes up, then output per 00:30:16.640 --> 00:30:19.280 worker will go up. 00:30:18.279 --> 00:30:21.240 Okay? 00:30:19.279 --> 00:30:22.960 The question is the next question is 00:30:21.240 --> 00:30:25.440 what happens to consumption per worker? 00:30:22.960 --> 00:30:27.519 Does consumption per worker go up 00:30:25.440 --> 00:30:29.559 or not? 00:30:27.519 --> 00:30:32.000 You are inclined to say, well, I mean, 00:30:29.559 --> 00:30:32.879 it makes sense that it goes up because 00:30:32.000 --> 00:30:34.759 uh 00:30:32.880 --> 00:30:38.080 we have more income, no? The saving rate 00:30:34.759 --> 00:30:41.079 is little s times y, then consumption is 00:30:38.079 --> 00:30:45.359 1 minus little s times y. So, income 00:30:41.079 --> 00:30:45.359 goes up, consumption should go up. 00:30:48.799 --> 00:30:54.519 And and yes, that's a dominant source, 00:30:52.359 --> 00:30:57.959 but it's not all the story because 00:30:54.519 --> 00:30:57.960 remember I I what I told you. 00:31:03.079 --> 00:31:05.639 So, consumption here is going to be 00:31:04.519 --> 00:31:08.759 equal to 00:31:05.640 --> 00:31:11.880 1 minus little s 00:31:08.759 --> 00:31:13.960 times y, so consumption 00:31:11.880 --> 00:31:15.280 per person will be 00:31:13.960 --> 00:31:17.559 that. 00:31:15.279 --> 00:31:20.000 Remember that what is increasing y over 00:31:17.559 --> 00:31:22.799 n there, so what is making this guy go 00:31:20.000 --> 00:31:24.440 up, which will lead to an increase in 00:31:22.799 --> 00:31:25.759 consumption over n, 00:31:24.440 --> 00:31:28.519 is that this 00:31:25.759 --> 00:31:28.519 guy went up. 00:31:28.640 --> 00:31:32.520 And that's a force in the opposite 00:31:30.079 --> 00:31:33.679 direction. 00:31:32.519 --> 00:31:36.200 Okay? 00:31:33.680 --> 00:31:38.080 So, in fact, that was one of the debates 00:31:36.200 --> 00:31:39.799 with the 00:31:38.079 --> 00:31:40.879 East Asian mirror Southeast Asian 00:31:39.799 --> 00:31:42.680 miracle 00:31:40.880 --> 00:31:44.160 is that it was fueled by lots of 00:31:42.680 --> 00:31:45.920 savings. So, people say, okay, that's 00:31:44.160 --> 00:31:47.600 wonderful. Your output growth is very 00:31:45.920 --> 00:31:49.600 fast, but consumption growth is not so 00:31:47.599 --> 00:31:51.039 fast. And at some point, it may be 00:31:49.599 --> 00:31:53.199 hurting you. I think that they were 00:31:51.039 --> 00:31:55.240 right though for other reasons, but 00:31:53.200 --> 00:31:58.000 but 00:31:55.240 --> 00:32:00.319 but that's that picture makes a point. 00:31:58.000 --> 00:32:02.480 You know, so if if if your saving rate 00:32:00.319 --> 00:32:04.519 to start with, this is a general lesson. 00:32:02.480 --> 00:32:07.079 If the saving rate is 00:32:04.519 --> 00:32:08.920 you start with is very very low, 00:32:07.079 --> 00:32:10.519 then an increase in the saving rate will 00:32:08.920 --> 00:32:12.640 lead to a strong increase in consumption 00:32:10.519 --> 00:32:14.759 because this change is a small relative 00:32:12.640 --> 00:32:16.880 to the big bang you get on output. 00:32:14.759 --> 00:32:19.679 Because if you have low saving rate, 00:32:16.880 --> 00:32:21.840 that also means that the 00:32:19.679 --> 00:32:24.200 the capital stock is very low. 00:32:21.839 --> 00:32:26.439 And if the capital stock is very low, f 00:32:24.200 --> 00:32:27.880 prime is is very big. You know, this is 00:32:26.440 --> 00:32:29.880 a concave function and you're in in the 00:32:27.880 --> 00:32:32.440 steep part of the function. 00:32:29.880 --> 00:32:34.080 Later on, if saving is very high, you're 00:32:32.440 --> 00:32:37.080 going to tend to have capital stock very 00:32:34.079 --> 00:32:39.599 high, and then first of all, 00:32:37.079 --> 00:32:41.480 more capital won't increase output per 00:32:39.599 --> 00:32:43.359 worker a lot because 00:32:41.480 --> 00:32:45.279 because of decreasing returns, 00:32:43.359 --> 00:32:46.519 and and this is a big number. So, it 00:32:45.279 --> 00:32:47.559 starts dominating. And that's what you 00:32:46.519 --> 00:32:50.879 see here. 00:32:47.559 --> 00:32:53.000 This economy as increases saving rate, 00:32:50.880 --> 00:32:55.440 uh consumption per worker rises, but at 00:32:53.000 --> 00:32:56.839 some point, it reaches a a maximum, and 00:32:55.440 --> 00:32:58.880 then it starts declining. 00:32:56.839 --> 00:33:01.480 I mean, think of the limit. If you save 00:32:58.880 --> 00:33:03.120 100% of your income, 00:33:01.480 --> 00:33:05.160 you don't consume anything. No matter 00:33:03.119 --> 00:33:07.839 how much is your output, if your saving 00:33:05.160 --> 00:33:09.720 rate is 100%, then you're not going to 00:33:07.839 --> 00:33:12.480 consume anything. 00:33:09.720 --> 00:33:15.039 If you have no income, no saving rate, 00:33:12.480 --> 00:33:16.839 no savings, no income, no capital stock, 00:33:15.039 --> 00:33:19.639 no income, you're not going to consume 00:33:16.839 --> 00:33:21.599 anything either. Okay? So, you at least 00:33:19.640 --> 00:33:23.080 you know these two points. And since you 00:33:21.599 --> 00:33:24.599 know there are some positive points in 00:33:23.079 --> 00:33:25.599 the in the middle, 00:33:24.599 --> 00:33:27.359 uh you know that the curve is going to 00:33:25.599 --> 00:33:28.519 tend to have that that kind of change. 00:33:27.359 --> 00:33:30.119 It's not going to be it's going to be 00:33:28.519 --> 00:33:33.160 non-monotonic. 00:33:30.119 --> 00:33:33.159 And that's the way 00:33:34.119 --> 00:33:37.759 So, let me just 00:33:36.079 --> 00:33:40.079 play with a little a few numbers. This 00:33:37.759 --> 00:33:40.079 is 00:33:40.400 --> 00:33:43.800 Yeah, let me play with a few numbers. 00:33:41.679 --> 00:33:45.679 It's not that crazy. 00:33:43.799 --> 00:33:47.559 Uh suppose you have a a production 00:33:45.679 --> 00:33:49.840 function that gives equal weight to 00:33:47.559 --> 00:33:51.639 capital and workers. So, this production 00:33:49.839 --> 00:33:53.279 function. 00:33:51.640 --> 00:33:55.800 That's a production function of constant 00:33:53.279 --> 00:33:57.119 return to scale. 00:33:55.799 --> 00:33:58.799 It better be because that's what we're 00:33:57.119 --> 00:34:01.359 doing, but 00:33:58.799 --> 00:34:01.359 what do you think? 00:34:01.720 --> 00:34:06.480 Yes, no. 00:34:03.039 --> 00:34:09.358 The sum of the exponents is one. So, 00:34:06.480 --> 00:34:11.000 it's k to the 1/2 n to the 1/2. The sum 00:34:09.358 --> 00:34:12.519 of the exponents is one, so you know 00:34:11.000 --> 00:34:14.239 that 00:34:12.519 --> 00:34:15.960 it's proportional to the scaling factor. 00:34:14.239 --> 00:34:17.479 So, 00:34:15.960 --> 00:34:20.878 we're going to use 00:34:17.480 --> 00:34:21.760 as a scaling as before n, so 00:34:20.878 --> 00:34:23.239 um 00:34:21.760 --> 00:34:24.520 so we have this. 00:34:23.239 --> 00:34:25.918 Okay? 00:34:24.519 --> 00:34:29.358 This is a this is a 00:34:25.918 --> 00:34:31.079 f of little f of k over n is the square 00:34:29.358 --> 00:34:32.239 root of k over n. 00:34:31.079 --> 00:34:34.319 Okay? 00:34:32.239 --> 00:34:37.119 Minus delta k over n. So, all that I'm 00:34:34.320 --> 00:34:39.240 doing is I'm plugging in that function. 00:34:37.119 --> 00:34:39.239 Uh 00:34:39.440 --> 00:34:44.039 So, here only 00:34:41.440 --> 00:34:45.039 I'm replacing all these functions by 00:34:44.039 --> 00:34:46.960 by a 00:34:45.039 --> 00:34:49.918 a specific example, one in which this is 00:34:46.960 --> 00:34:52.079 a square root of k over n. 00:34:49.918 --> 00:34:54.480 Okay? That's a concave function, square 00:34:52.079 --> 00:34:54.480 root. 00:34:56.398 --> 00:35:00.199 Good. 00:34:57.960 --> 00:35:01.679 Now, do it as an exercise. If you solve 00:35:00.199 --> 00:35:03.000 for the steady state, how do you solve 00:35:01.679 --> 00:35:04.759 for the steady state? Well, set this 00:35:03.000 --> 00:35:06.840 equal to zero. 00:35:04.760 --> 00:35:07.720 That will give you the steady state. 00:35:06.840 --> 00:35:09.440 No? 00:35:07.719 --> 00:35:11.119 If the steady state is when the capital 00:35:09.440 --> 00:35:13.039 is not growing anymore, it's when this 00:35:11.119 --> 00:35:14.960 is equal to zero. 00:35:13.039 --> 00:35:17.559 When this is equal to zero, I can solve 00:35:14.960 --> 00:35:19.079 for the steady state level of k over n, 00:35:17.559 --> 00:35:21.199 no, from here. 00:35:19.079 --> 00:35:22.519 This equal to zero, I can solve for k 00:35:21.199 --> 00:35:25.039 over n, and I'm going to call that the 00:35:22.519 --> 00:35:26.320 steady state. k star. 00:35:25.039 --> 00:35:29.000 We typically use the stars for the 00:35:26.320 --> 00:35:29.960 steady states in growth theory. 00:35:29.000 --> 00:35:33.480 Okay? 00:35:29.960 --> 00:35:35.840 Well, the answer to this is is k 00:35:33.480 --> 00:35:39.320 uh the steady state stock of capital per 00:35:35.840 --> 00:35:41.840 per person is the saving rate over delta 00:35:39.320 --> 00:35:43.280 squared. That's what it is. 00:35:41.840 --> 00:35:45.600 Output 00:35:43.280 --> 00:35:47.880 uh per person, which is the square root 00:35:45.599 --> 00:35:50.880 of k over n, is therefore the square 00:35:47.880 --> 00:35:52.240 root of s over delta squared, so it's s 00:35:50.880 --> 00:35:53.640 over delta. 00:35:52.239 --> 00:35:55.599 Okay? 00:35:53.639 --> 00:35:58.440 So, in this particular model, in the 00:35:55.599 --> 00:36:00.480 long run, output per worker doubles when 00:35:58.440 --> 00:36:03.079 the saving rate doubles. Okay? If I 00:36:00.480 --> 00:36:06.000 double the saving rate, then output per 00:36:03.079 --> 00:36:06.000 worker will double. 00:36:07.800 --> 00:36:11.880 Notice that the stock of capital is 00:36:09.760 --> 00:36:12.720 is going to grow a lot more 00:36:11.880 --> 00:36:14.960 in the 00:36:12.719 --> 00:36:15.559 when you increase the saving rate. 00:36:14.960 --> 00:36:18.199 Okay? 00:36:15.559 --> 00:36:18.199 It's square. 00:36:19.480 --> 00:36:23.440 So, in that economy, 00:36:21.159 --> 00:36:25.519 if you do increase the saving rate from 00:36:23.440 --> 00:36:27.320 10 to 20%, 00:36:25.519 --> 00:36:28.358 this is the way it goes. 00:36:27.320 --> 00:36:31.160 Okay? 00:36:28.358 --> 00:36:33.239 So, uh remember, 10 to 20% that means 00:36:31.159 --> 00:36:35.079 that the the new steady state output per 00:36:33.239 --> 00:36:36.759 worker will be twice what it was in the 00:36:35.079 --> 00:36:39.880 previous steady state. 00:36:36.760 --> 00:36:42.720 Okay? So, you go from one to two. 00:36:39.880 --> 00:36:44.760 But it takes a long time. 00:36:42.719 --> 00:36:48.319 And the numbers are not crazy. 50 years 00:36:44.760 --> 00:36:50.960 takes you to go to the new steady state. 00:36:48.320 --> 00:36:52.519 Okay? So, so that's sort of the time 00:36:50.960 --> 00:36:53.880 frame we're talking about. So, it is 00:36:52.519 --> 00:36:56.239 true that the saving rate will not 00:36:53.880 --> 00:37:00.039 change the long run rate of growth 00:36:56.239 --> 00:37:00.039 absent other mechanisms. 00:37:00.358 --> 00:37:04.039 But you can grow faster than your 00:37:02.400 --> 00:37:06.599 average, your steady state level for 00:37:04.039 --> 00:37:09.039 quite quite some time. Okay? And and 00:37:06.599 --> 00:37:13.480 again, a lot of that of the Asian 00:37:09.039 --> 00:37:13.480 miracle has been of that kind. 00:37:13.760 --> 00:37:17.800 This is what I was telling you of China 00:37:15.400 --> 00:37:19.358 before, no? Well, yeah, you you can grow 00:37:17.800 --> 00:37:22.120 very fast, especially if you have saving 00:37:19.358 --> 00:37:23.519 rate much higher than 20%, I mean, 50% 00:37:22.119 --> 00:37:25.719 or so. 00:37:23.519 --> 00:37:28.280 But but but the rate of growth will have 00:37:25.719 --> 00:37:29.719 a tendency to decline. Absent some other 00:37:28.280 --> 00:37:30.880 miracle, there are a lot of the reasons 00:37:29.719 --> 00:37:33.119 why we have all these fight about 00:37:30.880 --> 00:37:35.039 technology and so on. 00:37:33.119 --> 00:37:37.119 It has to do with cuz that's the main 00:37:35.039 --> 00:37:38.358 mechanism you alternative mechanism to 00:37:37.119 --> 00:37:40.400 grow. 00:37:38.358 --> 00:37:42.679 It's technology. Okay? We're going to 00:37:40.400 --> 00:37:44.960 talk about that in the next lecture. But 00:37:42.679 --> 00:37:47.159 but this force, which is what I'm saying 00:37:44.960 --> 00:37:49.960 the force, the easy part of growth, it's 00:37:47.159 --> 00:37:52.039 very difficult to fight this pattern. 00:37:49.960 --> 00:37:52.039 Okay? 00:37:53.480 --> 00:37:57.679 So, here you have numbers 00:37:55.840 --> 00:37:59.200 uh for the steady states. 00:37:57.679 --> 00:38:01.199 So, if the saving rate is zero, 00:37:59.199 --> 00:38:03.000 obviously, everything is zero. 00:38:01.199 --> 00:38:06.679 No way around. 00:38:03.000 --> 00:38:08.119 Uh if the saving rate is 0.1, 10%, then 00:38:06.679 --> 00:38:10.639 in this model, capital per worker is 00:38:08.119 --> 00:38:12.199 one, output per worker is one. 00:38:10.639 --> 00:38:13.559 Consumption per worker didn't go from 00:38:12.199 --> 00:38:15.480 zero to one. Why? Because you were 00:38:13.559 --> 00:38:18.079 saving something. So, it's zero is 1 00:38:15.480 --> 00:38:20.440 minus 0.1, which is the saving rate. 00:38:18.079 --> 00:38:22.159 Suppose you double the saving rate. 00:38:20.440 --> 00:38:23.760 Well, we know that we're going to double 00:38:22.159 --> 00:38:25.079 output per worker in this economy. We 00:38:23.760 --> 00:38:26.040 said that we're going to go from one to 00:38:25.079 --> 00:38:27.319 two. 00:38:26.039 --> 00:38:29.119 The capital stock is going to have to 00:38:27.320 --> 00:38:31.280 grow a lot more to double the amount of 00:38:29.119 --> 00:38:34.000 output. 00:38:31.280 --> 00:38:35.600 Why is that? Decreasing returns. 00:38:34.000 --> 00:38:37.239 To double output, you're going to have 00:38:35.599 --> 00:38:39.000 to much more than double capital 00:38:37.239 --> 00:38:42.039 because, you know, you need you're going 00:38:39.000 --> 00:38:44.320 to be fighting decreasing returns. 00:38:42.039 --> 00:38:46.239 What about uh consumption? Well, it 00:38:44.320 --> 00:38:48.039 won't double because you're doing this 00:38:46.239 --> 00:38:51.399 out of increasing the saving rate. So, 00:38:48.039 --> 00:38:52.320 you get the two minus now 0.2, not 0.1. 00:38:51.400 --> 00:38:56.320 Okay? 00:38:52.320 --> 00:38:58.000 Minus 0.2 times two. So, you get 1.6. 00:38:56.320 --> 00:38:59.039 And so on. 00:38:58.000 --> 00:39:01.840 And 00:38:59.039 --> 00:39:02.759 the higher you go with your saving rate, 00:39:01.840 --> 00:39:04.960 uh 00:39:02.760 --> 00:39:06.960 the harder it gets for capital to bring 00:39:04.960 --> 00:39:07.760 along uh 00:39:06.960 --> 00:39:09.199 uh 00:39:07.760 --> 00:39:11.040 um 00:39:09.199 --> 00:39:12.839 output per capita, 00:39:11.039 --> 00:39:14.759 and the more the drag on consumption 00:39:12.840 --> 00:39:17.358 because you need to be saving a lot in 00:39:14.760 --> 00:39:18.800 order to maintain this high stock of 00:39:17.358 --> 00:39:20.960 capital that you're having. Okay? You 00:39:18.800 --> 00:39:23.080 have a very large stock of capital, that 00:39:20.960 --> 00:39:25.199 means you need to save a lot just for 00:39:23.079 --> 00:39:28.279 the sake of maintaining that stock of 00:39:25.199 --> 00:39:30.039 capital. And so 00:39:28.280 --> 00:39:31.359 little is left for 00:39:30.039 --> 00:39:33.639 extra 00:39:31.358 --> 00:39:35.279 output per capita. And so, you see that 00:39:33.639 --> 00:39:37.400 here in this particular for this 00:39:35.280 --> 00:39:39.320 particular model, when the saving rate 00:39:37.400 --> 00:39:40.400 exceeds 0.5, 00:39:39.320 --> 00:39:42.559 then 00:39:40.400 --> 00:39:44.440 uh Uh, output obviously keeps rising 00:39:42.559 --> 00:39:46.320 when you increase the saving rate, but 00:39:44.440 --> 00:39:47.000 but output starts declining. So, that's 00:39:46.320 --> 00:39:48.640 your 00:39:47.000 --> 00:39:50.199 in the declining part. 00:39:48.639 --> 00:39:52.039 And if you get to one, of course, 00:39:50.199 --> 00:39:55.599 there's no consumption. So, that's a 00:39:52.039 --> 00:39:55.599 that's a curve that we trace. 00:39:59.639 --> 00:40:02.079 Okay. 00:40:03.599 --> 00:40:07.279 Is everything clear? Now, I'm going to 00:40:05.519 --> 00:40:10.039 That's a basic solo model, and that's a 00:40:07.280 --> 00:40:12.800 model that again you need to control 00:40:10.039 --> 00:40:14.559 completely. Okay. 00:40:12.800 --> 00:40:16.680 All that I'm going to do now is very 00:40:14.559 --> 00:40:19.000 simple. I'm going to just 00:40:16.679 --> 00:40:22.279 modify a little bit this model 00:40:19.000 --> 00:40:23.119 to uh add population growth. 00:40:22.280 --> 00:40:24.480 Okay. 00:40:23.119 --> 00:40:26.639 So, what happens 00:40:24.480 --> 00:40:28.719 By the way, 00:40:26.639 --> 00:40:32.000 for for centuries population growth has 00:40:28.719 --> 00:40:35.639 been one of the main In this model, 00:40:32.000 --> 00:40:37.920 we concluded that output per worker 00:40:35.639 --> 00:40:40.159 was not growing. 00:40:37.920 --> 00:40:41.639 What we're going to conclude in a second 00:40:40.159 --> 00:40:43.639 is that 00:40:41.639 --> 00:40:45.639 output per worker will not grow if 00:40:43.639 --> 00:40:48.359 population is growing. 00:40:45.639 --> 00:40:50.279 But that means that output is growing. 00:40:48.360 --> 00:40:52.760 If population is growing and output per 00:40:50.280 --> 00:40:54.880 worker is not growing, it's constant, 00:40:52.760 --> 00:40:56.640 that means output is also growing. And 00:40:54.880 --> 00:40:58.440 for a long time, 00:40:56.639 --> 00:41:01.159 growth 00:40:58.440 --> 00:41:04.119 of output, not of output per worker, was 00:41:01.159 --> 00:41:06.119 driven by large population growth. And 00:41:04.119 --> 00:41:07.519 sometimes you get big migration flows 00:41:06.119 --> 00:41:09.159 into a country that leads sort of to 00:41:07.519 --> 00:41:11.719 growth and so on. 00:41:09.159 --> 00:41:13.799 Now, big parts of the world 00:41:11.719 --> 00:41:15.719 have negative population growth. So, now 00:41:13.800 --> 00:41:17.120 we're going through a cycle in which is 00:41:15.719 --> 00:41:20.199 things are going the the other way 00:41:17.119 --> 00:41:22.079 around in in in many large parts of the 00:41:20.199 --> 00:41:24.399 world. I mean, this true in almost all 00:41:22.079 --> 00:41:26.679 of continental Europe, 00:41:24.400 --> 00:41:29.480 uh certainly in Japan, I said South 00:41:26.679 --> 00:41:31.599 Korea, China, 00:41:29.480 --> 00:41:34.079 and even some places Latin America. 00:41:31.599 --> 00:41:36.599 Okay. So, the drug actually is is 00:41:34.079 --> 00:41:38.319 against that. 00:41:36.599 --> 00:41:40.639 Uh we don't have the natural force for 00:41:38.320 --> 00:41:42.000 growth that we had for for many many 00:41:40.639 --> 00:41:44.400 years. 00:41:42.000 --> 00:41:46.320 So, let me let me introduce population 00:41:44.400 --> 00:41:47.840 growth. So, assume now that that 00:41:46.320 --> 00:41:50.280 population rather than being constant 00:41:47.840 --> 00:41:51.480 growth growth at the rate gn, which 00:41:50.280 --> 00:41:54.080 could be positive or negative. I'm going 00:41:51.480 --> 00:41:56.280 to do the example for the pos a positive 00:41:54.079 --> 00:41:58.880 uh population growth example. 00:41:56.280 --> 00:42:00.040 So, there's no equation that changes in 00:41:58.880 --> 00:42:02.240 the sense that 00:42:00.039 --> 00:42:05.039 this is still true. It's still true that 00:42:02.239 --> 00:42:09.119 investment equal to saving. It's still 00:42:05.039 --> 00:42:12.320 true that that uh output is equal to 00:42:09.119 --> 00:42:14.119 output per worker. Output is equal to f 00:42:12.320 --> 00:42:16.440 of k and n, 00:42:14.119 --> 00:42:17.960 and so on and so forth. 00:42:16.440 --> 00:42:20.720 The the thing that 00:42:17.960 --> 00:42:22.559 is a little trickier is that that, you 00:42:20.719 --> 00:42:24.079 know, 00:42:22.559 --> 00:42:27.679 in this model, 00:42:24.079 --> 00:42:30.039 if I don't normalize things for 00:42:27.679 --> 00:42:32.359 if I if I you know, in this case here 00:42:30.039 --> 00:42:34.480 where population was not growing, 00:42:32.360 --> 00:42:36.000 I could have just eliminated this n. 00:42:34.480 --> 00:42:37.840 It's a constant, and I would have done 00:42:36.000 --> 00:42:39.639 everything in in in capital in the space 00:42:37.840 --> 00:42:41.000 of capital here and output here. Would 00:42:39.639 --> 00:42:44.039 have been the same, just scaled by a 00:42:41.000 --> 00:42:46.159 number, a constant n. 00:42:44.039 --> 00:42:48.079 When I have population growth, I'm not 00:42:46.159 --> 00:42:49.239 indifferent between doing one way or the 00:42:48.079 --> 00:42:51.679 other. 00:42:49.239 --> 00:42:54.159 Because if I don't have if I don't if I 00:42:51.679 --> 00:42:56.079 do it in the space of k and y, 00:42:54.159 --> 00:42:58.119 and population is growing, then all 00:42:56.079 --> 00:42:59.599 these curves are moving. 00:42:58.119 --> 00:43:01.480 So, it's a very unfriendly diagram 00:42:59.599 --> 00:43:03.920 because my curves are all moving. As n 00:43:01.480 --> 00:43:05.519 is moving, everything is moving. So, the 00:43:03.920 --> 00:43:06.760 the trick in all these growth models, 00:43:05.519 --> 00:43:08.639 and it's going to be even more important 00:43:06.760 --> 00:43:11.120 in the next lecture, is to find the 00:43:08.639 --> 00:43:13.759 right scaling of capital so there is a 00:43:11.119 --> 00:43:16.759 steady state. So, you your curves are 00:43:13.760 --> 00:43:18.000 not moving around as population grows. 00:43:16.760 --> 00:43:20.440 It's very easy to find the scaling 00:43:18.000 --> 00:43:22.960 factor. It's population. 00:43:20.440 --> 00:43:25.400 Okay. So, 00:43:22.960 --> 00:43:27.480 that's what I'm going to do. 00:43:25.400 --> 00:43:29.519 But remember, what is different here is 00:43:27.480 --> 00:43:32.119 So, I want to what I'm saying here I 00:43:29.519 --> 00:43:33.880 want to get all my variables as scaled 00:43:32.119 --> 00:43:35.440 by population at some point in time. 00:43:33.880 --> 00:43:37.360 That's what I want to do. 00:43:35.440 --> 00:43:38.440 Because I know I practice enough with 00:43:37.360 --> 00:43:40.519 these things that's going to give me a 00:43:38.440 --> 00:43:41.200 steady state. Okay. 00:43:40.519 --> 00:43:43.519 Uh 00:43:41.199 --> 00:43:45.639 um Now, what is trickier relative to 00:43:43.519 --> 00:43:48.000 what I showed you before is that before 00:43:45.639 --> 00:43:49.759 I just divided by n both sides and and I 00:43:48.000 --> 00:43:51.960 was home. 00:43:49.760 --> 00:43:55.400 Now, I can't really do that. Okay. Let 00:43:51.960 --> 00:43:58.440 me divide by n t plus one both sides. 00:43:55.400 --> 00:44:01.400 So, that's nice. I get my capital per 00:43:58.440 --> 00:44:02.559 worker at t plus one. 00:44:01.400 --> 00:44:03.920 But there's certain things that are not 00:44:02.559 --> 00:44:05.079 as nice. 00:44:03.920 --> 00:44:06.599 What I have on the right-hand side is 00:44:05.079 --> 00:44:08.400 not what I really want. I don't want 00:44:06.599 --> 00:44:11.358 capital over 00:44:08.400 --> 00:44:12.960 population next period. 00:44:11.358 --> 00:44:14.799 Now, my steady state's going to be in 00:44:12.960 --> 00:44:17.000 the space of 00:44:14.800 --> 00:44:20.560 capital over population at the same 00:44:17.000 --> 00:44:23.000 time. That's my steady state. 00:44:20.559 --> 00:44:24.320 So, this is not so nice. 00:44:23.000 --> 00:44:26.440 So, what I have to do is I want to 00:44:24.320 --> 00:44:27.960 convert this the right-hand side in 00:44:26.440 --> 00:44:29.440 something that is of the kind of things 00:44:27.960 --> 00:44:31.039 that I want to have. 00:44:29.440 --> 00:44:33.679 So, what I'm going to do is divide and 00:44:31.039 --> 00:44:35.320 multiply each of these sides by nt over 00:44:33.679 --> 00:44:37.239 nt plus one. 00:44:35.320 --> 00:44:40.320 So, sorry. I'm going to divide and 00:44:37.239 --> 00:44:41.319 multiply each of these by nt. 00:44:40.320 --> 00:44:42.480 Okay. 00:44:41.320 --> 00:44:45.039 So, 00:44:42.480 --> 00:44:46.559 multiply by nt, divide by nt. So, I'm 00:44:45.039 --> 00:44:48.358 multiplying by one. 00:44:46.559 --> 00:44:50.119 Well, and and then I can rearrange the 00:44:48.358 --> 00:44:52.519 terms in this way. So, I get what I 00:44:50.119 --> 00:44:55.279 want, which is capital per 00:44:52.519 --> 00:44:58.199 uh person at time t, all at time t, but 00:44:55.280 --> 00:44:59.960 then I get this ratio here. 00:44:58.199 --> 00:45:01.879 Okay. And I can do the same for this 00:44:59.960 --> 00:45:05.280 expression here. 00:45:01.880 --> 00:45:05.280 Now, what is that ratio? 00:45:06.480 --> 00:45:12.119 Population 00:45:07.800 --> 00:45:12.120 today divided by population tomorrow. 00:45:15.119 --> 00:45:17.440 Well, 00:45:16.119 --> 00:45:19.480 it's one 00:45:17.440 --> 00:45:21.200 over one plus the rate of growth of 00:45:19.480 --> 00:45:23.559 population. 00:45:21.199 --> 00:45:25.279 nt plus one is equal to nt times one 00:45:23.559 --> 00:45:26.799 plus gn. 00:45:25.280 --> 00:45:29.600 That's the rate of growth 00:45:26.800 --> 00:45:29.600 of population. 00:45:41.760 --> 00:45:46.600 Okay. 00:45:43.679 --> 00:45:48.719 So, so what I have here 00:45:46.599 --> 00:45:52.079 is one over one plus g 00:45:48.719 --> 00:45:55.839 gn. Now, gn is not a big number. 00:45:52.079 --> 00:45:57.759 So, one over one plus gn, 00:45:55.840 --> 00:46:00.760 one over one plus gn is approximately 00:45:57.760 --> 00:46:02.920 equal to minus gn. 00:46:00.760 --> 00:46:05.120 Okay. So, one over one plus gn, gn is 00:46:02.920 --> 00:46:08.358 very close to zero, is approximately 00:46:05.119 --> 00:46:11.159 equal to minus gn. Okay. 00:46:08.358 --> 00:46:14.279 So, that's the reason this guy became 00:46:11.159 --> 00:46:16.559 that guy, approximately that guy. 00:46:14.280 --> 00:46:17.519 I can do the same here, but it turns out 00:46:16.559 --> 00:46:19.119 that 00:46:17.519 --> 00:46:20.440 the term there's an extra term here, 00:46:19.119 --> 00:46:22.920 therefore 00:46:20.440 --> 00:46:25.559 uh which is equal to 00:46:22.920 --> 00:46:27.920 s times gn 00:46:25.559 --> 00:46:29.039 times yt over nt. 00:46:27.920 --> 00:46:30.840 Well, that's second order. That's the 00:46:29.039 --> 00:46:33.279 reason I'm going to drop it. Okay. It's 00:46:30.840 --> 00:46:35.720 a saving rate, which is sorry it's it's 00:46:33.280 --> 00:46:36.519 a it's a small number times 00:46:35.719 --> 00:46:38.119 uh 00:46:36.519 --> 00:46:40.000 a rate of population growth, which is a 00:46:38.119 --> 00:46:41.679 number like, you know, 0.01 or something 00:46:40.000 --> 00:46:43.000 like that. So, that's a small number. 00:46:41.679 --> 00:46:44.639 So, I'm dropping it. 00:46:43.000 --> 00:46:46.679 That's a bigger approximation than that 00:46:44.639 --> 00:46:48.358 one, actually, but I'm going to do it. 00:46:46.679 --> 00:46:50.199 Everything becomes a lot simpler, but 00:46:48.358 --> 00:46:51.759 So, this is an approximation. 00:46:50.199 --> 00:46:54.119 Okay. I'm just dropping second-order 00:46:51.760 --> 00:46:54.120 terms. 00:46:54.559 --> 00:46:58.199 And once I have that, I have the system 00:46:56.280 --> 00:47:00.960 I want because now I have 00:46:58.199 --> 00:47:02.480 a a system for the evolution of the of 00:47:00.960 --> 00:47:04.440 the 00:47:02.480 --> 00:47:07.480 capital per 00:47:04.440 --> 00:47:10.200 po- per worker. Okay. 00:47:07.480 --> 00:47:12.079 Or per person. 00:47:10.199 --> 00:47:13.839 And if you see, it looks exactly as we 00:47:12.079 --> 00:47:16.079 had before. Remember, this is exactly 00:47:13.840 --> 00:47:18.960 what we had before. 00:47:16.079 --> 00:47:22.559 s f k over We used to have n not sub- 00:47:18.960 --> 00:47:24.440 subscript t. Now, it's k over nt. 00:47:22.559 --> 00:47:26.000 But what is different 00:47:24.440 --> 00:47:27.960 is that now, rather than having only the 00:47:26.000 --> 00:47:29.400 depreciation rate here, we have the 00:47:27.960 --> 00:47:32.400 depreciation rate plus the rate of 00:47:29.400 --> 00:47:32.400 growth of population. 00:47:32.719 --> 00:47:36.719 Why do you think we have the rate of 00:47:33.840 --> 00:47:39.358 growth of population there? 00:47:36.719 --> 00:47:42.319 Remember the the the economics 00:47:39.358 --> 00:47:44.599 behind this expression before. 00:47:42.320 --> 00:47:47.840 It was 00:47:44.599 --> 00:47:49.920 This is what adds to capital. 00:47:47.840 --> 00:47:52.160 To capital per worker. 00:47:49.920 --> 00:47:54.240 This is what you need to maintain. What 00:47:52.159 --> 00:47:55.399 takes away from capital. 00:47:54.239 --> 00:47:56.639 Okay. 00:47:55.400 --> 00:47:58.400 Now, 00:47:56.639 --> 00:47:59.719 it's what takes away from 00:47:58.400 --> 00:48:01.240 given we're doing everything in the 00:47:59.719 --> 00:48:04.119 space of capital per worker, that takes 00:48:01.239 --> 00:48:05.759 away from capital Oh, that's a typo. 00:48:04.119 --> 00:48:07.599 There's a t there. 00:48:05.760 --> 00:48:09.880 Okay. 00:48:07.599 --> 00:48:09.880 t 00:48:11.358 --> 00:48:14.319 Okay. 00:48:12.800 --> 00:48:16.120 So, why do you think I have this gn 00:48:14.320 --> 00:48:17.760 here? 00:48:16.119 --> 00:48:19.400 Well, I have only one minute, so I don't 00:48:17.760 --> 00:48:21.960 have time to. 00:48:19.400 --> 00:48:23.519 Because if I want to maintain a stock of 00:48:21.960 --> 00:48:25.119 capital 00:48:23.519 --> 00:48:26.519 per worker, 00:48:25.119 --> 00:48:28.039 and workers 00:48:26.519 --> 00:48:29.920 are growing, 00:48:28.039 --> 00:48:31.920 then I need to be growing the capital 00:48:29.920 --> 00:48:33.559 stock. Even if I had no depreciation, if 00:48:31.920 --> 00:48:35.840 I want to maintain the capital per 00:48:33.559 --> 00:48:37.039 worker constant, and workers are 00:48:35.840 --> 00:48:38.120 growing, 00:48:37.039 --> 00:48:39.639 then I need to grow the stock of 00:48:38.119 --> 00:48:41.599 capital. 00:48:39.639 --> 00:48:43.679 So, in order to maintain the capital I 00:48:41.599 --> 00:48:46.279 still need to spend what I used to spend 00:48:43.679 --> 00:48:48.199 for depreciation of the capital stock. 00:48:46.280 --> 00:48:50.920 But if I want to maintain the the 00:48:48.199 --> 00:48:52.839 capital per worker constant, then I'm 00:48:50.920 --> 00:48:53.840 going to need more investment. 00:48:52.840 --> 00:48:56.559 Okay. 00:48:53.840 --> 00:48:58.160 Just to make make up for that that extra 00:48:56.559 --> 00:49:01.559 component. 00:48:58.159 --> 00:49:03.199 So, now, set ga equal to zero. That's 00:49:01.559 --> 00:49:05.599 Your diagram is exactly as before in 00:49:03.199 --> 00:49:06.559 this space. Set a equal to one and 00:49:05.599 --> 00:49:07.759 constant. 00:49:06.559 --> 00:49:10.358 But this 00:49:07.760 --> 00:49:15.200 line, the red line here, will have delta 00:49:10.358 --> 00:49:15.199 plus gn. Okay. So, it rotates up. 00:49:17.119 --> 00:49:20.199 So, you can play here and see what 00:49:18.800 --> 00:49:21.320 happens if there's change in population 00:49:20.199 --> 00:49:23.199 growth, 00:49:21.320 --> 00:49:24.359 and so on and so forth. 00:49:23.199 --> 00:49:26.519 It's going to be counterintuitive 00:49:24.358 --> 00:49:28.358 initially because you see, if I increase 00:49:26.519 --> 00:49:29.840 population growth, this curve will 00:49:28.358 --> 00:49:31.840 rotate up, 00:49:29.840 --> 00:49:35.000 and then it would appear as if that 00:49:31.840 --> 00:49:35.000 leads to negative growth. 00:49:35.760 --> 00:49:40.240 But you don't get negative growth. In 00:49:37.400 --> 00:49:43.920 this diagram, you do get that 00:49:40.239 --> 00:49:44.879 Y over over N will decline. 00:49:43.920 --> 00:49:47.119 But that doesn't mean that you get 00:49:44.880 --> 00:49:48.480 negative growth. It just means that 00:49:47.119 --> 00:49:50.599 output is not growing as fast as 00:49:48.480 --> 00:49:52.199 population. 00:49:50.599 --> 00:49:53.279 But but both are growing. Just the 00:49:52.199 --> 00:49:56.839 population is growing faster than 00:49:53.280 --> 00:49:57.400 output. I'll I'll I'll start from that. 00:49:56.840 --> 00:49:59.079 Uh 00:49:57.400 --> 00:50:00.440 oh, I think it's after your break. So, 00:49:59.079 --> 00:50:02.480 you're going to have forgotten 00:50:00.440 --> 00:50:04.840 everything by then. So, I'll do a review 00:50:02.480 --> 00:50:08.400 of this and then and then we 00:50:04.840 --> 00:50:08.400 Okay. Have a Have a nice break.