WEBVTT 00:00:00.000 --> 00:00:02.440 align:middle line:90% [SQUEAKING] 00:00:02.440 --> 00:00:03.904 align:middle line:90% [RUSTLING] 00:00:03.904 --> 00:00:05.368 align:middle line:90% [CLICKING] 00:00:05.368 --> 00:00:16.455 align:middle line:90% 00:00:16.455 --> 00:00:18.080 align:middle line:84% RICARDEO CABALLERO: I couldn't connect. 00:00:18.080 --> 00:00:21.960 align:middle line:84% So the Fed just hiked by 25 basis points. 00:00:21.960 --> 00:00:25.370 align:middle line:90% And as people expected-- 00:00:25.370 --> 00:00:26.630 align:middle line:90% this is the way it works. 00:00:26.630 --> 00:00:28.005 align:middle line:84% When there's lots of uncertainty, 00:00:28.005 --> 00:00:31.400 align:middle line:84% essentially, the Fed starts communicating 00:00:31.400 --> 00:00:33.980 align:middle line:84% what it's going to do, and the communication 00:00:33.980 --> 00:00:40.100 align:middle line:84% was very clear that 25 basis points was to be expected. 00:00:40.100 --> 00:00:42.200 align:middle line:84% And apparently-- I was reading this right now. 00:00:42.200 --> 00:00:46.760 align:middle line:84% It was released 3 minutes ago, 4 minutes ago-- 00:00:46.760 --> 00:00:52.140 align:middle line:84% they also said that further hikes are no longer guaranteed. 00:00:52.140 --> 00:00:56.570 align:middle line:84% So remember that we saw that expected hikes-- 00:00:56.570 --> 00:00:59.870 align:middle line:84% we saw several expected hikes for the next few months 00:00:59.870 --> 00:01:02.990 align:middle line:90% before the SVB mess. 00:01:02.990 --> 00:01:06.380 align:middle line:84% And right after it, we saw the whole thing declining. 00:01:06.380 --> 00:01:11.150 align:middle line:84% And at least the minutes are consistent with that. 00:01:11.150 --> 00:01:12.530 align:middle line:90% So there we are. 00:01:12.530 --> 00:01:13.523 align:middle line:90% So not big uncertainty. 00:01:13.523 --> 00:01:15.940 align:middle line:84% I imagine the markets are rallying or something like that, 00:01:15.940 --> 00:01:17.610 align:middle line:84% at least for the next 10 minutes or so. 00:01:17.610 --> 00:01:20.790 align:middle line:90% But we shall see. 00:01:20.790 --> 00:01:25.320 align:middle line:84% Anyway, but today, we're going to really start-- 00:01:25.320 --> 00:01:29.700 align:middle line:84% I'm going to show you the first model of economic growth. 00:01:29.700 --> 00:01:33.915 align:middle line:84% And before I do that, who knows who that person is? 00:01:33.915 --> 00:01:37.740 align:middle line:90% 00:01:37.740 --> 00:01:39.000 align:middle line:90% No, no clue? 00:01:39.000 --> 00:01:42.780 align:middle line:90% 00:01:42.780 --> 00:01:46.020 align:middle line:90% Actually, he's Robert Solow. 00:01:46.020 --> 00:01:48.210 align:middle line:84% He's an emeritus professor at MIT. 00:01:48.210 --> 00:01:50.310 align:middle line:84% Together with Paul Samuelson, essentially, he's 00:01:50.310 --> 00:01:53.760 align:middle line:84% responsible for building the Economics Department at MIT. 00:01:53.760 --> 00:01:57.030 align:middle line:84% And he won the Nobel Prize in 1987-- 00:01:57.030 --> 00:01:58.320 align:middle line:90% I was a student then here-- 00:01:58.320 --> 00:02:03.990 align:middle line:90% 00:02:03.990 --> 00:02:06.370 align:middle line:84% primarily for his work on economic growth. 00:02:06.370 --> 00:02:10.169 align:middle line:84% And so what we're going to do in the next two or three lectures 00:02:10.169 --> 00:02:15.570 align:middle line:84% are essentially things that Bob Solow developed many, many years 00:02:15.570 --> 00:02:17.610 align:middle line:90% ago. 00:02:17.610 --> 00:02:20.130 align:middle line:84% The basic mechanism-- you remember 00:02:20.130 --> 00:02:22.050 align:middle line:84% that we had this Keynesian cross before where 00:02:22.050 --> 00:02:24.265 align:middle line:84% we had this multiplier in the goods market 00:02:24.265 --> 00:02:26.640 align:middle line:84% and aggregate demand feeding into income and so on and so 00:02:26.640 --> 00:02:27.160 align:middle line:90% forth. 00:02:27.160 --> 00:02:31.350 align:middle line:84% That was the star mechanism in short-run macro. 00:02:31.350 --> 00:02:37.480 align:middle line:84% In long-run macro growth theory, this is the key mechanism, 00:02:37.480 --> 00:02:39.400 align:middle line:84% and you can think of it as the following. 00:02:39.400 --> 00:02:43.110 align:middle line:84% At any point in time, an economy has 00:02:43.110 --> 00:02:47.140 align:middle line:84% factors of production-- primarily, labor and capital. 00:02:47.140 --> 00:02:48.640 align:middle line:90% That capital is stock. 00:02:48.640 --> 00:02:50.140 align:middle line:84% Labor is more or less fixed, so it 00:02:50.140 --> 00:02:51.700 align:middle line:84% depends on population, growth, things 00:02:51.700 --> 00:02:54.970 align:middle line:84% that are difficult to control, or they're not 00:02:54.970 --> 00:02:58.780 align:middle line:84% really that endogenous to economics, not, at least, 00:02:58.780 --> 00:03:00.610 align:middle line:90% in the current times. 00:03:00.610 --> 00:03:02.530 align:middle line:84% Many centuries ago, yes, they were. 00:03:02.530 --> 00:03:06.160 align:middle line:84% We had this Malthusian theories, in which population growth 00:03:06.160 --> 00:03:10.240 align:middle line:84% determined growth because food scarcity and stuff like that. 00:03:10.240 --> 00:03:14.530 align:middle line:84% But that's no longer the case, fortunately, 00:03:14.530 --> 00:03:16.636 align:middle line:90% in most parts of the world. 00:03:16.636 --> 00:03:20.560 align:middle line:84% But what can change over time and quite a bit, 00:03:20.560 --> 00:03:23.770 align:middle line:84% and it depends on economic decisions, is the capital stock. 00:03:23.770 --> 00:03:26.540 align:middle line:84% But at any point in time, there is certain capital stocks, 00:03:26.540 --> 00:03:29.620 align:middle line:84% which, combined with labor, give you some certain output. 00:03:29.620 --> 00:03:31.300 align:middle line:90% Output is income. 00:03:31.300 --> 00:03:34.390 align:middle line:84% Part of that income will be saved, as we have seen. 00:03:34.390 --> 00:03:39.660 align:middle line:84% And those savings will be used for investment. 00:03:39.660 --> 00:03:43.450 align:middle line:84% But investment is nothing else than capital accumulation. 00:03:43.450 --> 00:03:46.960 align:middle line:84% So this income will lead to savings, 00:03:46.960 --> 00:03:48.690 align:middle line:84% which will fund investment, which 00:03:48.690 --> 00:03:50.070 align:middle line:84% will change the stock of capital, 00:03:50.070 --> 00:03:51.862 align:middle line:84% will feed into the capital stock, that will 00:03:51.862 --> 00:03:53.530 align:middle line:90% feed into income, and so on. 00:03:53.530 --> 00:03:56.280 align:middle line:84% All this is happening very slowly because the capital 00:03:56.280 --> 00:03:57.480 align:middle line:90% stock accumulates slowly. 00:03:57.480 --> 00:04:00.120 align:middle line:90% 00:04:00.120 --> 00:04:01.595 align:middle line:90% But this is what is happening. 00:04:01.595 --> 00:04:02.970 align:middle line:84% And so all the models we're going 00:04:02.970 --> 00:04:04.845 align:middle line:84% to look at-- certainly, the model we're going 00:04:04.845 --> 00:04:07.800 align:middle line:84% to look at in this lecture is all about this mechanism. 00:04:07.800 --> 00:04:10.460 align:middle line:90% 00:04:10.460 --> 00:04:14.005 align:middle line:84% So let's remember what we did in the previous lecture. 00:04:14.005 --> 00:04:16.579 align:middle line:90% 00:04:16.579 --> 00:04:18.920 align:middle line:84% And I'm going to assume that population is constant. 00:04:18.920 --> 00:04:20.300 align:middle line:84% I'm going to relax that at the very end. 00:04:20.300 --> 00:04:22.730 align:middle line:84% But assume that the population is constant and equal to N. 00:04:22.730 --> 00:04:25.370 align:middle line:84% And remember, we're not worried about unemployment and stuff 00:04:25.370 --> 00:04:27.090 align:middle line:90% like that here. 00:04:27.090 --> 00:04:33.860 align:middle line:84% And so output per capita or per person is Y over N. 00:04:33.860 --> 00:04:36.680 align:middle line:84% And we remember we had a production function 00:04:36.680 --> 00:04:40.490 align:middle line:84% f of K and N. Then because of constant returns to scale, 00:04:40.490 --> 00:04:43.940 align:middle line:84% we could divide by N on both sides, everything. 00:04:43.940 --> 00:04:47.520 align:middle line:84% And we ended up with this relationship. 00:04:47.520 --> 00:04:51.060 align:middle line:84% So output per person is equal to-- 00:04:51.060 --> 00:04:54.170 align:middle line:84% is an increasing function of capital per person. 00:04:54.170 --> 00:04:56.330 align:middle line:84% It's an increasing function of capital per person, 00:04:56.330 --> 00:05:01.070 align:middle line:84% but it's also a concave function of capital per person. 00:05:01.070 --> 00:05:02.660 align:middle line:90% Why is it concave? 00:05:02.660 --> 00:05:05.360 align:middle line:84% That is, why is it increasing at a smaller pace? 00:05:05.360 --> 00:05:14.240 align:middle line:90% 00:05:14.240 --> 00:05:15.212 align:middle line:90% Yeah? 00:05:15.212 --> 00:05:17.160 align:middle line:84% AUDIENCE: It's decreasing marginal method. 00:05:17.160 --> 00:05:19.770 align:middle line:84% RICARDEO CABALLERO: Decreasing marginal product 00:05:19.770 --> 00:05:21.990 align:middle line:90% of capital, exactly. 00:05:21.990 --> 00:05:23.850 align:middle line:84% For fixed amount of labor, the more capital 00:05:23.850 --> 00:05:28.440 align:middle line:84% you put into production, well, output keeps expanding, 00:05:28.440 --> 00:05:31.350 align:middle line:84% but by less and less because it has less and less labor 00:05:31.350 --> 00:05:33.990 align:middle line:84% to work with, each unit of capital. 00:05:33.990 --> 00:05:34.740 align:middle line:90% Perfect. 00:05:34.740 --> 00:05:36.960 align:middle line:90% That's very important. 00:05:36.960 --> 00:05:39.510 align:middle line:84% Then we're going to work in closed economy. 00:05:39.510 --> 00:05:40.380 align:middle line:90% I haven't opened it. 00:05:40.380 --> 00:05:43.800 align:middle line:84% I'm going to do that after quiz 2. 00:05:43.800 --> 00:05:46.410 align:middle line:90% 00:05:46.410 --> 00:05:49.300 align:middle line:84% And I'm going to assume, also, no public deficits-- so g 00:05:49.300 --> 00:05:52.630 align:middle line:84% equal to T, capital T. And in that case, 00:05:52.630 --> 00:05:57.220 align:middle line:84% then, we know that private investment-- private savings 00:05:57.220 --> 00:05:58.920 align:middle line:90% equal to private investment. 00:05:58.920 --> 00:06:02.340 align:middle line:84% That's the way we derived our S curve. 00:06:02.340 --> 00:06:06.040 align:middle line:90% So that's not new. 00:06:06.040 --> 00:06:10.310 align:middle line:84% I'm going to modify a little bit what we did in the short run, 00:06:10.310 --> 00:06:12.510 align:middle line:84% and I'm going to assume that savings 00:06:12.510 --> 00:06:14.110 align:middle line:90% is proportional to income. 00:06:14.110 --> 00:06:18.690 align:middle line:84% So savings-- little s times Y. Notice 00:06:18.690 --> 00:06:22.870 align:middle line:84% that this is different from what we did in the short run. 00:06:22.870 --> 00:06:25.290 align:middle line:84% In the short run, remember, we had a C0 floating around. 00:06:25.290 --> 00:06:28.120 align:middle line:84% We had a constant in the consumption function. 00:06:28.120 --> 00:06:31.050 align:middle line:84% So savings, which was equal to income minus consumption, 00:06:31.050 --> 00:06:33.930 align:middle line:84% also had a constant floating around. 00:06:33.930 --> 00:06:36.810 align:middle line:84% Now, that constant was important in the short-run model 00:06:36.810 --> 00:06:39.570 align:middle line:84% because we were approximating for a bunch of things that are 00:06:39.570 --> 00:06:41.400 align:middle line:84% not related to short-term income-- 00:06:41.400 --> 00:06:46.500 align:middle line:84% wealth, the price of houses, and stuff like that. 00:06:46.500 --> 00:06:49.860 align:middle line:84% We put all that in that constant there. 00:06:49.860 --> 00:06:52.710 align:middle line:84% When you think about the long run, though, 00:06:52.710 --> 00:06:55.500 align:middle line:84% most of those things that we excluded there-- 00:06:55.500 --> 00:06:57.450 align:middle line:90% asset prices, stuff like that-- 00:06:57.450 --> 00:07:00.540 align:middle line:84% tend to scale with output, as well. 00:07:00.540 --> 00:07:03.550 align:middle line:84% So these are inconsistent on the surface. 00:07:03.550 --> 00:07:05.100 align:middle line:84% But if you were to fully work out 00:07:05.100 --> 00:07:08.640 align:middle line:84% what is behind the C0 in the consumption function, 00:07:08.640 --> 00:07:11.280 align:middle line:84% then this is not a bad approximation. 00:07:11.280 --> 00:07:13.200 align:middle line:84% They are not that inconsistent because you 00:07:13.200 --> 00:07:17.100 align:middle line:84% endogenize things that, over the long run, scale with income. 00:07:17.100 --> 00:07:20.080 align:middle line:84% I mean, wealth tends to rise with income, 00:07:20.080 --> 00:07:22.480 align:middle line:84% and all these things tend to move together-- not 00:07:22.480 --> 00:07:23.380 align:middle line:90% at high frequency. 00:07:23.380 --> 00:07:25.120 align:middle line:84% You can have all sorts of fluctuations. 00:07:25.120 --> 00:07:28.580 align:middle line:84% But over the long run, they tend to scale up together. 00:07:28.580 --> 00:07:31.220 align:middle line:84% So that's going to be our saving function. 00:07:31.220 --> 00:07:35.660 align:middle line:84% So that means that we know, in equilibrium-- 00:07:35.660 --> 00:07:37.360 align:middle line:90% this is not investment function. 00:07:37.360 --> 00:07:42.160 align:middle line:84% We know that, in equilibrium, investment will be equal to-- 00:07:42.160 --> 00:07:45.270 align:middle line:84% it will be proportional to income. 00:07:45.270 --> 00:07:48.900 align:middle line:84% So remember where we were going through the box. 00:07:48.900 --> 00:07:51.150 align:middle line:84% At the top of the box, we had capital. 00:07:51.150 --> 00:07:52.800 align:middle line:90% That led to output. 00:07:52.800 --> 00:07:54.450 align:middle line:84% We're doing everything per capita. 00:07:54.450 --> 00:08:01.560 align:middle line:84% That led to savings, and that funded investment. 00:08:01.560 --> 00:08:03.700 align:middle line:90% So that's what we have. 00:08:03.700 --> 00:08:10.080 align:middle line:84% So this growth model is really about these three functional 00:08:10.080 --> 00:08:11.880 align:middle line:84% forms and then a dynamic equation 00:08:11.880 --> 00:08:13.540 align:middle line:90% for the stock of capital. 00:08:13.540 --> 00:08:18.070 align:middle line:84% So the evolution of the stock of capital-- 00:08:18.070 --> 00:08:21.240 align:middle line:84% capital will increase because of investment. 00:08:21.240 --> 00:08:22.590 align:middle line:90% That's what investment is. 00:08:22.590 --> 00:08:26.250 align:middle line:84% It's an increase in the stock of capital. 00:08:26.250 --> 00:08:30.360 align:middle line:84% But it will also decrease as a result of depreciation. 00:08:30.360 --> 00:08:36.043 align:middle line:84% I mean, things do break up once in a while. 00:08:36.043 --> 00:08:38.460 align:middle line:84% And different types of capital have different depreciation 00:08:38.460 --> 00:08:39.120 align:middle line:90% rates. 00:08:39.120 --> 00:08:41.610 align:middle line:84% Equipment depreciates much faster than structures, 00:08:41.610 --> 00:08:43.600 align:middle line:90% and buildings, and so on. 00:08:43.600 --> 00:08:46.170 align:middle line:84% But we're not going to make those distinctions here. 00:08:46.170 --> 00:08:49.350 align:middle line:84% But you see this tells you, the capital stock at t plus 1 00:08:49.350 --> 00:08:52.740 align:middle line:84% is equal to the capital stock we had before minus 00:08:52.740 --> 00:08:55.080 align:middle line:84% what is depreciated of that stock of capital 00:08:55.080 --> 00:08:57.510 align:middle line:84% plus any new investment we do today. 00:08:57.510 --> 00:09:00.410 align:middle line:90% 00:09:00.410 --> 00:09:02.690 align:middle line:84% In per worker terms-- and remember that, for now, 00:09:02.690 --> 00:09:05.300 align:middle line:84% I'm keeping population growth constant-- 00:09:05.300 --> 00:09:07.480 align:middle line:84% equal to 0-- not population growth 00:09:07.480 --> 00:09:09.230 align:middle line:84% constant-- yeah, constant, but equal to 0. 00:09:09.230 --> 00:09:12.110 align:middle line:90% So population is constant. 00:09:12.110 --> 00:09:15.140 align:middle line:84% I can divide these both sides by N, 00:09:15.140 --> 00:09:18.710 align:middle line:84% and I get that capital per worker. 00:09:18.710 --> 00:09:23.690 align:middle line:84% Per worker or per person is equal to this expression here. 00:09:23.690 --> 00:09:24.650 align:middle line:90% I did two things here. 00:09:24.650 --> 00:09:29.420 align:middle line:84% I divided by N, and I replaced this I function, 00:09:29.420 --> 00:09:33.837 align:middle line:84% this investment, for savings because I 00:09:33.837 --> 00:09:35.670 align:middle line:84% know, in equilibrium, they have to be equal. 00:09:35.670 --> 00:09:38.420 align:middle line:90% So I have that. 00:09:38.420 --> 00:09:39.740 align:middle line:90% I can rewrite this. 00:09:39.740 --> 00:09:43.400 align:middle line:84% Just subtract Kt over N on both sides, 00:09:43.400 --> 00:09:46.400 align:middle line:84% and then you get-- the change in capital per person 00:09:46.400 --> 00:09:50.570 align:middle line:84% is an increasing function of savings 00:09:50.570 --> 00:09:53.060 align:middle line:90% and decreasing of depreciation. 00:09:53.060 --> 00:09:58.310 align:middle line:84% So the last step that is important in this model is to-- 00:09:58.310 --> 00:10:01.070 align:middle line:84% so here, I have, essentially, a difference equation for capital, 00:10:01.070 --> 00:10:04.040 align:middle line:84% but we have an output per capita on the right-hand side. 00:10:04.040 --> 00:10:07.890 align:middle line:84% But it turns out that I know that output per capita, 00:10:07.890 --> 00:10:09.050 align:middle line:90% per person-- 00:10:09.050 --> 00:10:10.432 align:middle line:90% I said per capita, per person. 00:10:10.432 --> 00:10:11.390 align:middle line:90% They're the same thing. 00:10:11.390 --> 00:10:15.330 align:middle line:84% Per worker-- it's the same thing in this part of the course. 00:10:15.330 --> 00:10:19.286 align:middle line:90% So this is equal to-- 00:10:19.286 --> 00:10:21.740 align:middle line:84% is a function, is an increasing and concave function 00:10:21.740 --> 00:10:25.290 align:middle line:90% of capital per person. 00:10:25.290 --> 00:10:28.530 align:middle line:84% So this is-- I would say it's a fundamental equation 00:10:28.530 --> 00:10:30.270 align:middle line:90% of the Solow growth model. 00:10:30.270 --> 00:10:35.370 align:middle line:84% It says, the change in the stock of capital increases 00:10:35.370 --> 00:10:42.600 align:middle line:84% with investment, of course, and decreases with depreciation. 00:10:42.600 --> 00:10:45.390 align:middle line:84% And both of these expressions here 00:10:45.390 --> 00:10:51.630 align:middle line:84% are increasing functions of the stock of capital per person. 00:10:51.630 --> 00:10:54.670 align:middle line:84% So let's try to understand what is in here. 00:10:54.670 --> 00:11:01.470 align:middle line:84% So why-- so this is linear, obviously, 00:11:01.470 --> 00:11:03.900 align:middle line:84% because the depreciation is linear. 00:11:03.900 --> 00:11:07.140 align:middle line:84% Say you lose 5% of your stock of capital every year 00:11:07.140 --> 00:11:09.180 align:middle line:90% because it breaks down. 00:11:09.180 --> 00:11:12.190 align:middle line:84% Obviously, the more capital per person you have, 00:11:12.190 --> 00:11:14.580 align:middle line:84% the more units of capital you're going to lose. 00:11:14.580 --> 00:11:16.620 align:middle line:84% This is units of capital per person. 00:11:16.620 --> 00:11:19.470 align:middle line:84% You have a larger stock of capital, you're going to lose-- 00:11:19.470 --> 00:11:22.688 align:middle line:84% 5% of a larger number is a larger number. 00:11:22.688 --> 00:11:23.730 align:middle line:90% And this is proportional. 00:11:23.730 --> 00:11:24.870 align:middle line:90% It's linear. 00:11:24.870 --> 00:11:27.240 align:middle line:84% Now, this one-- remember, this comes here 00:11:27.240 --> 00:11:29.460 align:middle line:90% from the saving function. 00:11:29.460 --> 00:11:33.850 align:middle line:84% And this term here is equal to income per person. 00:11:33.850 --> 00:11:36.780 align:middle line:84% Now, suppose that you start in a situation 00:11:36.780 --> 00:11:39.340 align:middle line:84% where the capital stock is relatively low, 00:11:39.340 --> 00:11:42.500 align:middle line:90% and this is positive. 00:11:42.500 --> 00:11:44.350 align:middle line:84% What does it mean that this is positive? 00:11:44.350 --> 00:11:46.570 align:middle line:84% I mean, the implication of this being positive 00:11:46.570 --> 00:11:50.420 align:middle line:84% is that the stock of capital per person will be growing. 00:11:50.420 --> 00:11:52.855 align:middle line:84% But what does it mean that it's positive in words? 00:11:52.855 --> 00:11:56.190 align:middle line:90% 00:11:56.190 --> 00:11:58.450 align:middle line:84% I mean, you have a stock of capital. 00:11:58.450 --> 00:12:00.840 align:middle line:84% There are things that reduce the stock of capital, 00:12:00.840 --> 00:12:03.490 align:middle line:84% and there are things that increase the stock of capital. 00:12:03.490 --> 00:12:06.250 align:middle line:84% This is the thing that increases the stock of capital. 00:12:06.250 --> 00:12:08.770 align:middle line:84% That's the thing that reduces the stock of capital. 00:12:08.770 --> 00:12:17.000 align:middle line:84% So if this is greater than that, what does that mean? 00:12:17.000 --> 00:12:18.220 align:middle line:90% That means it is positive. 00:12:18.220 --> 00:12:20.590 align:middle line:90% But in words, what is happening? 00:12:20.590 --> 00:12:31.360 align:middle line:90% 00:12:31.360 --> 00:12:33.300 align:middle line:84% And we simplify, but remember, this 00:12:33.300 --> 00:12:34.920 align:middle line:90% is just investment per person. 00:12:34.920 --> 00:12:39.240 align:middle line:90% 00:12:39.240 --> 00:12:43.980 align:middle line:84% Well, this just says that in this economy, 00:12:43.980 --> 00:12:46.740 align:middle line:84% there is more investment than destruction 00:12:46.740 --> 00:12:49.850 align:middle line:90% of capital due to depreciation. 00:12:49.850 --> 00:12:51.260 align:middle line:90% That's what this means. 00:12:51.260 --> 00:12:53.220 align:middle line:90% This is investment. 00:12:53.220 --> 00:12:56.540 align:middle line:84% And this is positive means that the investment, which 00:12:56.540 --> 00:12:58.940 align:middle line:84% is a function of saving, the saving rate, and stuff 00:12:58.940 --> 00:13:00.290 align:middle line:90% like that-- 00:13:00.290 --> 00:13:03.530 align:middle line:84% it's a function of the funding available for investment-- 00:13:03.530 --> 00:13:07.640 align:middle line:84% it's equal to the funding available for investment. 00:13:07.640 --> 00:13:10.010 align:middle line:84% If this is positive, well, this is greater 00:13:10.010 --> 00:13:12.200 align:middle line:90% than the stock of capital. 00:13:12.200 --> 00:13:14.840 align:middle line:84% Another way of saying it-- you need a minimum level 00:13:14.840 --> 00:13:17.915 align:middle line:84% of investment in an economy to maintain the stock of capital. 00:13:17.915 --> 00:13:20.282 align:middle line:90% 00:13:20.282 --> 00:13:21.740 align:middle line:84% The minimum level of investment you 00:13:21.740 --> 00:13:23.750 align:middle line:84% need to maintain the stock of capital 00:13:23.750 --> 00:13:25.410 align:middle line:90% is equal to the depreciation. 00:13:25.410 --> 00:13:30.050 align:middle line:84% So 10 machine breaks, you need to invest at least 10 machines 00:13:30.050 --> 00:13:34.190 align:middle line:84% in order to maintain the stock of capital constant. 00:13:34.190 --> 00:13:35.658 align:middle line:84% Now if this is positive, it means 00:13:35.658 --> 00:13:37.700 align:middle line:84% that you're investing more than the machines that 00:13:37.700 --> 00:13:39.080 align:middle line:90% are breaking down. 00:13:39.080 --> 00:13:42.780 align:middle line:84% Now, suppose you start in a situation where that's the case. 00:13:42.780 --> 00:13:45.740 align:middle line:84% So that means the stock of capital is growing. 00:13:45.740 --> 00:13:47.720 align:middle line:84% Suppose I ask you, the next period, 00:13:47.720 --> 00:13:51.350 align:middle line:84% do you think the gap will be larger or smaller than it 00:13:51.350 --> 00:13:52.070 align:middle line:90% used to be? 00:13:52.070 --> 00:14:02.220 align:middle line:90% 00:14:02.220 --> 00:14:05.400 align:middle line:84% Actually, that's not a great question 00:14:05.400 --> 00:14:10.480 align:middle line:84% because I'm not doing it in the right units for that. 00:14:10.480 --> 00:14:14.310 align:middle line:84% Let me ask you a variation of that question. 00:14:14.310 --> 00:14:16.800 align:middle line:90% Suppose we keep going. 00:14:16.800 --> 00:14:19.860 align:middle line:84% After a while, do you think that number 00:14:19.860 --> 00:14:23.670 align:middle line:90% will get larger or smaller? 00:14:23.670 --> 00:14:26.460 align:middle line:90% Let it run for quite a while. 00:14:26.460 --> 00:14:29.400 align:middle line:84% Do you think that number will-- so remember what I'm saying. 00:14:29.400 --> 00:14:31.180 align:middle line:84% We start from some stock of capital. 00:14:31.180 --> 00:14:32.333 align:middle line:90% This is positive. 00:14:32.333 --> 00:14:34.500 align:middle line:84% If this is positive, it means that the capital stock 00:14:34.500 --> 00:14:35.260 align:middle line:90% is growing. 00:14:35.260 --> 00:14:38.460 align:middle line:84% That means this guy is growing, and that guy is growing, 00:14:38.460 --> 00:14:40.860 align:middle line:90% and they're growing equally. 00:14:40.860 --> 00:14:43.230 align:middle line:84% But after a while, do you think this number will 00:14:43.230 --> 00:14:45.750 align:middle line:90% get smaller or bigger-- 00:14:45.750 --> 00:14:47.790 align:middle line:84% after a long while, just to make sure 00:14:47.790 --> 00:14:49.520 align:middle line:84% that my approximation is not tidy. 00:14:49.520 --> 00:14:55.448 align:middle line:90% 00:14:55.448 --> 00:14:57.930 align:middle line:84% AUDIENCE: Smaller because of the decrease in [INAUDIBLE].. 00:14:57.930 --> 00:14:59.097 align:middle line:90% RICARDEO CABALLERO: Exactly. 00:14:59.097 --> 00:15:01.170 align:middle line:84% It's going to get smaller because this guy keeps 00:15:01.170 --> 00:15:05.370 align:middle line:84% growing linearly [INAUDIBLE] and this one is not. 00:15:05.370 --> 00:15:06.840 align:middle line:90% It's concave. 00:15:06.840 --> 00:15:08.850 align:middle line:90% At some point, this income-- 00:15:08.850 --> 00:15:10.800 align:middle line:84% you need to put a lot of capital for income 00:15:10.800 --> 00:15:13.050 align:middle line:84% to keep rising, and therefore, for saving 00:15:13.050 --> 00:15:15.900 align:middle line:84% to keep rising, and therefore, for investment to keep rising. 00:15:15.900 --> 00:15:20.378 align:middle line:84% And at some point, yes, it won't be able to really grow. 00:15:20.378 --> 00:15:22.920 align:middle line:84% I mean, you're going to be using all your investment, really, 00:15:22.920 --> 00:15:25.350 align:middle line:84% to maintain the stock of capital. 00:15:25.350 --> 00:15:30.000 align:middle line:84% That's the logic of the Solow model. 00:15:30.000 --> 00:15:31.532 align:middle line:90% And it's all in this diagram. 00:15:31.532 --> 00:15:33.990 align:middle line:84% So this is the diagram you should really, really understand 00:15:33.990 --> 00:15:37.830 align:middle line:84% well and control it, and play with it, and all that. 00:15:37.830 --> 00:15:40.110 align:middle line:84% It's the equivalent to your IS-LM 00:15:40.110 --> 00:15:44.440 align:middle line:84% model in the first part of the course. 00:15:44.440 --> 00:15:46.760 align:middle line:90% So look at what we have here. 00:15:46.760 --> 00:15:51.070 align:middle line:84% So I'm going to plot output per worker-- 00:15:51.070 --> 00:15:52.570 align:middle line:90% per worker, per person-- 00:15:52.570 --> 00:15:56.030 align:middle line:90% against capital per worker here. 00:15:56.030 --> 00:16:04.570 align:middle line:84% And so this red line here is just the depreciation, 00:16:04.570 --> 00:16:09.550 align:middle line:84% this term here, and thus, is a linear function 00:16:09.550 --> 00:16:12.160 align:middle line:90% of the capital per worker. 00:16:12.160 --> 00:16:15.270 align:middle line:90% That's what it is. 00:16:15.270 --> 00:16:20.890 align:middle line:84% The blue line here is output per worker, which, as we said, 00:16:20.890 --> 00:16:24.420 align:middle line:84% is a concave function of K over N. Remember, 00:16:24.420 --> 00:16:26.420 align:middle line:84% I showed you that production function 00:16:26.420 --> 00:16:28.300 align:middle line:90% in the previous lecture. 00:16:28.300 --> 00:16:30.150 align:middle line:90% There you are. 00:16:30.150 --> 00:16:32.130 align:middle line:90% What is the green line? 00:16:32.130 --> 00:16:34.230 align:middle line:84% It's investment per worker, which 00:16:34.230 --> 00:16:35.910 align:middle line:90% is equal to saving per worker. 00:16:35.910 --> 00:16:39.420 align:middle line:84% And saving per worker is little s, the saving rate, 00:16:39.420 --> 00:16:41.890 align:middle line:90% times a output. 00:16:41.890 --> 00:16:44.910 align:middle line:84% So it's little s, which is a number like 0.1, 00:16:44.910 --> 00:16:49.320 align:middle line:84% if we're talking about the US, and 0.4 if we're 00:16:49.320 --> 00:16:50.670 align:middle line:90% talking about Singapore. 00:16:50.670 --> 00:16:52.395 align:middle line:84% It varies a lot across countries. 00:16:52.395 --> 00:16:55.440 align:middle line:90% 00:16:55.440 --> 00:16:58.080 align:middle line:84% So this green line here is nothing else 00:16:58.080 --> 00:17:01.342 align:middle line:84% than this blue line multiplied by a number that is less than 1. 00:17:01.342 --> 00:17:02.550 align:middle line:90% That's the reason it's lower. 00:17:02.550 --> 00:17:06.140 align:middle line:90% 00:17:06.140 --> 00:17:07.010 align:middle line:90% OK, good. 00:17:07.010 --> 00:17:12.270 align:middle line:84% So the point I was describing before was a point like this. 00:17:12.270 --> 00:17:13.579 align:middle line:90% Remember? 00:17:13.579 --> 00:17:15.800 align:middle line:90% The point I was describing-- 00:17:15.800 --> 00:17:19.099 align:middle line:84% suppose that the economy starts in a point like this one, K0 00:17:19.099 --> 00:17:24.250 align:middle line:84% over N. And I want to understand the dynamics of this economy. 00:17:24.250 --> 00:17:26.480 align:middle line:90% How will it grow over time? 00:17:26.480 --> 00:17:33.090 align:middle line:84% So what you see here is that, at this level 00:17:33.090 --> 00:17:37.030 align:middle line:84% of capital per worker, investment 00:17:37.030 --> 00:17:41.390 align:middle line:90% is greater than depreciation. 00:17:41.390 --> 00:17:43.690 align:middle line:84% So that's exactly a situation where 00:17:43.690 --> 00:17:50.585 align:middle line:84% this is positive That distance here is that. 00:17:50.585 --> 00:17:53.980 align:middle line:90% 00:17:53.980 --> 00:17:56.160 align:middle line:84% And the reason I said, ah, I'm not 00:17:56.160 --> 00:17:58.660 align:middle line:84% going to do any local analysis because we could have started 00:17:58.660 --> 00:18:02.660 align:middle line:84% with a K over 0 over here, and then that number is growing. 00:18:02.660 --> 00:18:04.630 align:middle line:84% But it's growing-- if you were to normalize 00:18:04.630 --> 00:18:07.645 align:middle line:84% by the stock of capital, it's declining. 00:18:07.645 --> 00:18:09.320 align:middle line:84% But I didn't want to do that then. 00:18:09.320 --> 00:18:12.380 align:middle line:84% But now, that's what I-- so let's look at this case. 00:18:12.380 --> 00:18:14.320 align:middle line:84% You're in a situation where this is positive. 00:18:14.320 --> 00:18:18.610 align:middle line:84% If this is positive, it means the capital stock per worker 00:18:18.610 --> 00:18:19.330 align:middle line:90% is growing. 00:18:19.330 --> 00:18:21.540 align:middle line:90% So you're moving to the right. 00:18:21.540 --> 00:18:25.110 align:middle line:84% In the next period, you're going to be here. 00:18:25.110 --> 00:18:26.610 align:middle line:84% That means the capital-- so it keeps 00:18:26.610 --> 00:18:30.990 align:middle line:90% growing but by smaller steps. 00:18:30.990 --> 00:18:35.170 align:middle line:84% Eventually, the investment is entirely 00:18:35.170 --> 00:18:41.270 align:middle line:84% used for recovering from the depreciation of capital, 00:18:41.270 --> 00:18:43.490 align:middle line:84% so covering the depreciation of capital. 00:18:43.490 --> 00:18:47.830 align:middle line:84% And at that point, the capital stock stops growing. 00:18:47.830 --> 00:18:50.440 align:middle line:84% We call that a steady state, stationary state. 00:18:50.440 --> 00:18:53.190 align:middle line:90% We stop there. 00:18:53.190 --> 00:18:55.500 align:middle line:84% So that's the steady state of this model. 00:18:55.500 --> 00:18:58.892 align:middle line:84% That means this economy, regardless of where I-- 00:18:58.892 --> 00:19:00.600 align:middle line:84% I'll do the analysis from the other side. 00:19:00.600 --> 00:19:02.683 align:middle line:84% Suppose that you start from a situation like this. 00:19:02.683 --> 00:19:05.250 align:middle line:90% You start with a lot of capital. 00:19:05.250 --> 00:19:08.730 align:middle line:84% Well, if you start with a lot of capital in this economy, 00:19:08.730 --> 00:19:12.100 align:middle line:90% what happens here? 00:19:12.100 --> 00:19:14.520 align:middle line:84% Well, what happens here is that the investment 00:19:14.520 --> 00:19:16.620 align:middle line:84% you're putting into the ground in this economy 00:19:16.620 --> 00:19:20.030 align:middle line:84% is less than what you need to maintain the stock of capital, 00:19:20.030 --> 00:19:21.640 align:middle line:90% which is the depreciation. 00:19:21.640 --> 00:19:23.200 align:middle line:84% And that means the stock of capital 00:19:23.200 --> 00:19:26.050 align:middle line:90% will be shrinking over time. 00:19:26.050 --> 00:19:27.440 align:middle line:90% You're moving that way. 00:19:27.440 --> 00:19:31.180 align:middle line:84% So regardless of where you start in this economy, 00:19:31.180 --> 00:19:34.150 align:middle line:84% if I ask you the question A hundred years from now, 00:19:34.150 --> 00:19:36.998 align:middle line:90% where are you? 00:19:36.998 --> 00:19:39.290 align:middle line:84% you tell me, I don't need to know where you start from. 00:19:39.290 --> 00:19:41.350 align:middle line:84% I know that we're going to end up around there. 00:19:41.350 --> 00:19:44.350 align:middle line:84% You can-- either you start from here, you can go there. 00:19:44.350 --> 00:19:46.063 align:middle line:84% From here, you go there and so on. 00:19:46.063 --> 00:19:47.980 align:middle line:84% That's the reason we call this a steady state. 00:19:47.980 --> 00:19:50.185 align:middle line:84% This is where you converge in the long run. 00:19:50.185 --> 00:19:54.940 align:middle line:90% 00:19:54.940 --> 00:20:00.390 align:middle line:84% Now, this is already interesting because it tells you, 00:20:00.390 --> 00:20:04.060 align:middle line:84% at this point here, the economy was growing. 00:20:04.060 --> 00:20:08.560 align:middle line:84% The capital stock was growing, and output was growing. 00:20:08.560 --> 00:20:09.700 align:middle line:90% You see the capital-- 00:20:09.700 --> 00:20:12.880 align:middle line:84% if you start from here, the capital stock is growing. 00:20:12.880 --> 00:20:15.400 align:middle line:90% Well, output is also growing. 00:20:15.400 --> 00:20:18.000 align:middle line:90% You're moving up. 00:20:18.000 --> 00:20:19.770 align:middle line:90% So you had growth. 00:20:19.770 --> 00:20:24.000 align:middle line:84% That kind of growth we call transitional growth. 00:20:24.000 --> 00:20:27.660 align:middle line:84% It goes from one point to another point. 00:20:27.660 --> 00:20:28.980 align:middle line:90% It's not a permanent growth. 00:20:28.980 --> 00:20:30.520 align:middle line:90% It's a transitional growth. 00:20:30.520 --> 00:20:33.480 align:middle line:84% It's the fact that you were away from your steady state, 00:20:33.480 --> 00:20:37.450 align:middle line:84% and then you're going convergent towards your steady state. 00:20:37.450 --> 00:20:40.210 align:middle line:84% A lot of the growth we observed and the difference of growth we 00:20:40.210 --> 00:20:41.500 align:middle line:84% observed across countries-- remember, 00:20:41.500 --> 00:20:43.583 align:middle line:84% I showed you those downward-sloping curves and all 00:20:43.583 --> 00:20:44.290 align:middle line:90% that-- 00:20:44.290 --> 00:20:46.570 align:middle line:90% is as a result of that. 00:20:46.570 --> 00:20:50.242 align:middle line:84% Poorer economies tend to have lower capital-- 00:20:50.242 --> 00:20:53.320 align:middle line:84% capital labor, capital employment ratios, 00:20:53.320 --> 00:20:54.940 align:middle line:90% capital population ratios. 00:20:54.940 --> 00:20:58.180 align:middle line:84% And therefore, they tend to grow faster because they're catching 00:20:58.180 --> 00:20:59.920 align:middle line:90% up with the steady state. 00:20:59.920 --> 00:21:02.350 align:middle line:84% Very advanced economies that have been more or less 00:21:02.350 --> 00:21:05.500 align:middle line:84% in the same place for a long time are moving around there, 00:21:05.500 --> 00:21:08.260 align:middle line:84% so there's less catching up growth. 00:21:08.260 --> 00:21:11.050 align:middle line:84% And that's the main responsible for the downward-sloping 00:21:11.050 --> 00:21:14.680 align:middle line:84% curve I showed you within OECD countries and even broader 00:21:14.680 --> 00:21:15.250 align:middle line:90% than that. 00:21:15.250 --> 00:21:18.920 align:middle line:84% Africa was a little bit of a problem there. 00:21:18.920 --> 00:21:23.550 align:middle line:84% OK, so this is an important model 00:21:23.550 --> 00:21:26.050 align:middle line:90% for you, important diagram. 00:21:26.050 --> 00:21:28.300 align:middle line:90% Let's play a little with it. 00:21:28.300 --> 00:21:31.690 align:middle line:84% So suppose that, at the time-- this is a very simple model. 00:21:31.690 --> 00:21:37.470 align:middle line:84% But at the time, the view was that, well, 00:21:37.470 --> 00:21:40.390 align:middle line:84% what really supports growth is savings. 00:21:40.390 --> 00:21:43.500 align:middle line:84% So economies that save a lot grow a lot. 00:21:43.500 --> 00:21:45.450 align:middle line:84% And this sort of makes sense here 00:21:45.450 --> 00:21:48.460 align:middle line:84% because investment, which is what 00:21:48.460 --> 00:21:51.250 align:middle line:84% leads to capital accumulation, is entirely funded by savings. 00:21:51.250 --> 00:21:52.630 align:middle line:90% It makes sense. 00:21:52.630 --> 00:21:55.900 align:middle line:84% You have more savings, you should grow more. 00:21:55.900 --> 00:21:59.060 align:middle line:84% OK, so this is something-- we can do an experiment. 00:21:59.060 --> 00:22:03.220 align:middle line:84% Suppose you start at a steady state, if you will. 00:22:03.220 --> 00:22:06.600 align:middle line:84% And now we increase the savings rate. 00:22:06.600 --> 00:22:08.910 align:middle line:90% What moves? 00:22:08.910 --> 00:22:09.722 align:middle line:90% Which curve? 00:22:09.722 --> 00:22:11.430 align:middle line:84% This is the kind of thing you should know 00:22:11.430 --> 00:22:13.680 align:middle line:90% when you work with this model. 00:22:13.680 --> 00:22:18.960 align:middle line:84% If I change the saving rate, which curve moves in this model? 00:22:18.960 --> 00:22:20.760 align:middle line:90% Let me go one by one. 00:22:20.760 --> 00:22:22.140 align:middle line:90% Does the red line move? 00:22:22.140 --> 00:22:24.800 align:middle line:90% 00:22:24.800 --> 00:22:25.300 align:middle line:90% No. 00:22:25.300 --> 00:22:26.560 align:middle line:84% It has nothing to do with savings. 00:22:26.560 --> 00:22:27.852 align:middle line:90% It has to do with depreciation. 00:22:27.852 --> 00:22:32.320 align:middle line:84% If I move depreciation rate, that curve will move, but not-- 00:22:32.320 --> 00:22:35.070 align:middle line:84% will the production function move? 00:22:35.070 --> 00:22:35.790 align:middle line:90% No. 00:22:35.790 --> 00:22:37.590 align:middle line:90% So the blue line cannot move. 00:22:37.590 --> 00:22:39.420 align:middle line:84% All that will move is the green line 00:22:39.420 --> 00:22:45.310 align:middle line:84% because the green line is the saving rate times the blue line. 00:22:45.310 --> 00:22:46.770 align:middle line:84% So if I increase the saving rate, 00:22:46.770 --> 00:22:49.620 align:middle line:84% I'm going to move the green line up. 00:22:49.620 --> 00:22:52.390 align:middle line:90% That's what we have here. 00:22:52.390 --> 00:22:55.500 align:middle line:84% So you see what happens is you start-- 00:22:55.500 --> 00:22:59.100 align:middle line:84% this was a steady state for this saving rate in this economy. 00:22:59.100 --> 00:23:02.760 align:middle line:84% Now, all of a sudden, this economy starts saving more. 00:23:02.760 --> 00:23:05.040 align:middle line:90% What happens? 00:23:05.040 --> 00:23:07.500 align:middle line:84% This tells you very much the story of Asia. 00:23:07.500 --> 00:23:12.000 align:middle line:84% The Asia miracle of the '60s, '70s, and so on is very much 00:23:12.000 --> 00:23:14.530 align:middle line:90% something like that-- 00:23:14.530 --> 00:23:18.280 align:middle line:90% a little more complicated. 00:23:18.280 --> 00:23:23.800 align:middle line:84% A big part of what explains the fast growth of Asia 00:23:23.800 --> 00:23:27.943 align:middle line:84% during that period is that something like that happened. 00:23:27.943 --> 00:23:29.860 align:middle line:84% Now, why the savings rate increase and so on-- 00:23:29.860 --> 00:23:31.443 align:middle line:84% that's all very interesting and so on. 00:23:31.443 --> 00:23:34.460 align:middle line:84% But it's not what I want to discuss today. 00:23:34.460 --> 00:23:37.000 align:middle line:90% So what happened here, then? 00:23:37.000 --> 00:23:39.402 align:middle line:84% So what happens-- this economy was in a steady state. 00:23:39.402 --> 00:23:40.360 align:middle line:90% So there was no growth. 00:23:40.360 --> 00:23:46.220 align:middle line:84% It was growing at 0 in a steady state because this says, 00:23:46.220 --> 00:23:49.950 align:middle line:84% in a steady state, output per worker remains constant. 00:23:49.950 --> 00:23:52.040 align:middle line:84% And since we have no population growth, 00:23:52.040 --> 00:23:54.620 align:middle line:84% that means output is not growing, either. 00:23:54.620 --> 00:23:56.120 align:middle line:84% The only way you can have that ratio 00:23:56.120 --> 00:23:58.850 align:middle line:84% constant with the denominator not moving 00:23:58.850 --> 00:24:02.330 align:middle line:84% is that the numerator is not moving, either. 00:24:02.330 --> 00:24:03.500 align:middle line:90% OK, good. 00:24:03.500 --> 00:24:08.270 align:middle line:84% So now, boom, all of a sudden, we get a higher saving rate. 00:24:08.270 --> 00:24:11.010 align:middle line:90% So what happens now? 00:24:11.010 --> 00:24:11.595 align:middle line:90% What reacts? 00:24:11.595 --> 00:24:14.260 align:middle line:90% 00:24:14.260 --> 00:24:15.900 align:middle line:90% So the saving rates go up. 00:24:15.900 --> 00:24:17.080 align:middle line:90% It's a closed economy. 00:24:17.080 --> 00:24:20.880 align:middle line:84% It means the investment rate will go up. 00:24:20.880 --> 00:24:21.990 align:middle line:90% What happens now? 00:24:21.990 --> 00:24:29.670 align:middle line:90% 00:24:29.670 --> 00:24:31.800 align:middle line:90% What does that gap tell you? 00:24:31.800 --> 00:24:35.010 align:middle line:90% 00:24:35.010 --> 00:24:37.360 align:middle line:84% Now you have a positive gap there, 00:24:37.360 --> 00:24:41.280 align:middle line:84% which means you're investing more than what you need in order 00:24:41.280 --> 00:24:44.890 align:middle line:84% to maintain the stock of capital at the previous steady state. 00:24:44.890 --> 00:24:46.645 align:middle line:84% So that means that the stock of capital 00:24:46.645 --> 00:24:48.270 align:middle line:84% is going to start growing to the right. 00:24:48.270 --> 00:24:50.460 align:middle line:90% It's going to start growing. 00:24:50.460 --> 00:24:54.900 align:middle line:84% And as the stock of capital grows, then output per capita 00:24:54.900 --> 00:24:56.560 align:middle line:90% also grows. 00:24:56.560 --> 00:24:58.500 align:middle line:84% And this will keep happening until you 00:24:58.500 --> 00:25:02.030 align:middle line:90% reach the new steady state. 00:25:02.030 --> 00:25:05.970 align:middle line:84% So a higher saving rate-- so important conclusion there. 00:25:05.970 --> 00:25:11.120 align:middle line:84% This, as simple as it is, proves something-- 00:25:11.120 --> 00:25:14.690 align:middle line:84% that the conventional wisdom that the higher saving 00:25:14.690 --> 00:25:18.080 align:middle line:84% rate would give you sustained growth, higher growth, 00:25:18.080 --> 00:25:21.470 align:middle line:84% isn't really true, certainly not in this model. 00:25:21.470 --> 00:25:24.680 align:middle line:84% Eventually, you'll go back to growth equal to 0. 00:25:24.680 --> 00:25:26.720 align:middle line:84% When you reach a new steady state, 00:25:26.720 --> 00:25:28.370 align:middle line:84% you're going to be also growing at 0. 00:25:28.370 --> 00:25:31.310 align:middle line:90% 00:25:31.310 --> 00:25:34.370 align:middle line:84% What is true, though, is that you get, again, 00:25:34.370 --> 00:25:38.990 align:middle line:84% what is called transitional growth because here, you're 00:25:38.990 --> 00:25:41.570 align:middle line:84% going to start growing very fast, in fact. 00:25:41.570 --> 00:25:43.430 align:middle line:84% And then you're going to keep growing 00:25:43.430 --> 00:25:45.920 align:middle line:84% at the lowest-- lower, lower pace until you go back to 0. 00:25:45.920 --> 00:25:47.503 align:middle line:84% But you're going to get lots of growth 00:25:47.503 --> 00:25:49.920 align:middle line:84% in the transition as a result of that. 00:25:49.920 --> 00:25:51.500 align:middle line:84% And it turns out, in the data, when 00:25:51.500 --> 00:25:53.510 align:middle line:84% you're looking at 20, 30 years of data, 00:25:53.510 --> 00:25:59.030 align:middle line:84% it's difficult to disentangle very permanent rates of growth 00:25:59.030 --> 00:26:01.320 align:middle line:84% versus transitional rate of growth. 00:26:01.320 --> 00:26:04.130 align:middle line:84% This is one of the things that has concerned China quite a bit. 00:26:04.130 --> 00:26:05.910 align:middle line:90% They grow very, very fast. 00:26:05.910 --> 00:26:08.280 align:middle line:84% They have been growing very, very fast for a long time. 00:26:08.280 --> 00:26:10.800 align:middle line:84% But it's very clear that it's becoming harder and harder 00:26:10.800 --> 00:26:12.780 align:middle line:84% for them to grow at the type of rates of growth 00:26:12.780 --> 00:26:16.110 align:middle line:90% that they had 20 years ago. 00:26:16.110 --> 00:26:18.720 align:middle line:84% They had rates of growth 15% or so. 00:26:18.720 --> 00:26:22.380 align:middle line:84% They had very high-- they had a very low initial capital 00:26:22.380 --> 00:26:24.120 align:middle line:90% population ratio-- 00:26:24.120 --> 00:26:29.350 align:middle line:84% big population, little capital, and enormous savings rates. 00:26:29.350 --> 00:26:32.230 align:middle line:90% So they grew very, very fast. 00:26:32.230 --> 00:26:34.020 align:middle line:84% They had the green line very close 00:26:34.020 --> 00:26:37.470 align:middle line:84% to the blue line, the capital stock very low, 00:26:37.470 --> 00:26:40.808 align:middle line:84% so they grew very, very fast but they have been growing very fast 00:26:40.808 --> 00:26:42.100 align:middle line:90% for a very long period of time. 00:26:42.100 --> 00:26:44.100 align:middle line:84% So now it's getting a lot harder because they're 00:26:44.100 --> 00:26:46.230 align:middle line:84% getting closer and closer to their steady state. 00:26:46.230 --> 00:26:47.790 align:middle line:90% That's the issue. 00:26:47.790 --> 00:26:49.410 align:middle line:84% There are other sources of growth, 00:26:49.410 --> 00:26:52.470 align:middle line:84% and that's what we're going to talk about in the next lecture. 00:26:52.470 --> 00:26:56.783 align:middle line:84% But this is sometimes called the easy part of growth. 00:26:56.783 --> 00:26:58.200 align:middle line:84% It's sort of running out in China. 00:26:58.200 --> 00:27:09.780 align:middle line:90% 00:27:09.780 --> 00:27:12.780 align:middle line:84% And it has run out in all developed economies 00:27:12.780 --> 00:27:13.710 align:middle line:90% for quite a while. 00:27:13.710 --> 00:27:18.950 align:middle line:90% 00:27:18.950 --> 00:27:19.450 align:middle line:90% Good. 00:27:19.450 --> 00:27:22.180 align:middle line:90% 00:27:22.180 --> 00:27:23.950 align:middle line:90% Is this clear? 00:27:23.950 --> 00:27:24.630 align:middle line:90% It's important. 00:27:24.630 --> 00:27:29.940 align:middle line:84% I mean, a question like that is guaranteed in your quiz for 81. 00:27:29.940 --> 00:27:32.438 align:middle line:84% What happens if the saving rate does something? 00:27:32.438 --> 00:27:35.190 align:middle line:90% 00:27:35.190 --> 00:27:38.650 align:middle line:90% So this is a plot over time. 00:27:38.650 --> 00:27:41.070 align:middle line:84% So this is a case in which you were in a steady state. 00:27:41.070 --> 00:27:44.580 align:middle line:84% And at time t, the saving rate goes up 00:27:44.580 --> 00:27:47.610 align:middle line:90% as 1 greater than 0 jump. 00:27:47.610 --> 00:27:50.430 align:middle line:90% Then output cannot jump. 00:27:50.430 --> 00:27:53.845 align:middle line:84% So the saving rate goes up, but output cannot jump at day 0. 00:27:53.845 --> 00:27:54.345 align:middle line:90% Why? 00:27:54.345 --> 00:27:56.430 align:middle line:84% Why is it that the output doesn't jump immediately 00:27:56.430 --> 00:27:57.360 align:middle line:90% to a new steady state? 00:27:57.360 --> 00:28:02.900 align:middle line:90% 00:28:02.900 --> 00:28:04.250 align:middle line:90% This is the-- 00:28:04.250 --> 00:28:07.250 align:middle line:84% I'm saying, this is what will happen to output. 00:28:07.250 --> 00:28:09.620 align:middle line:84% You're going to start growing very fast early on, 00:28:09.620 --> 00:28:11.180 align:middle line:84% and then keep growing, keep growing 00:28:11.180 --> 00:28:13.790 align:middle line:84% at a slower and lower pace because of decreasing returns 00:28:13.790 --> 00:28:14.960 align:middle line:90% to capital. 00:28:14.960 --> 00:28:18.020 align:middle line:84% And eventually, you'll convert to the new steady state 00:28:18.020 --> 00:28:22.640 align:middle line:84% with the rate of growth equal to 0, like the one you had before, 00:28:22.640 --> 00:28:25.120 align:middle line:90% this savings shock. 00:28:25.120 --> 00:28:28.210 align:middle line:84% And the question I'm asking now is, why does output 00:28:28.210 --> 00:28:29.120 align:middle line:90% have to do this? 00:28:29.120 --> 00:28:30.220 align:middle line:90% Why doesn't it just jump? 00:28:30.220 --> 00:28:35.300 align:middle line:90% 00:28:35.300 --> 00:28:37.895 align:middle line:84% What is the only variable that could make it jump? 00:28:37.895 --> 00:28:41.490 align:middle line:90% 00:28:41.490 --> 00:28:43.465 align:middle line:84% So you need to look at the production function. 00:28:43.465 --> 00:28:45.090 align:middle line:84% The production function is the function 00:28:45.090 --> 00:28:46.860 align:middle line:90% of K over N. N is fixed. 00:28:46.860 --> 00:28:49.830 align:middle line:84% The only thing that can make it jump is the capital stock jumps, 00:28:49.830 --> 00:28:52.020 align:middle line:84% but the capital stock is not jumping. 00:28:52.020 --> 00:28:53.550 align:middle line:90% That's a stock. 00:28:53.550 --> 00:28:55.020 align:middle line:84% And in order to accumulate a larger 00:28:55.020 --> 00:28:56.760 align:middle line:84% stock of the new steady state, you're 00:28:56.760 --> 00:28:58.650 align:middle line:84% going to go through a lot of flows. 00:28:58.650 --> 00:28:59.825 align:middle line:90% That's investment. 00:28:59.825 --> 00:29:01.950 align:middle line:84% Every year, you're going to be adding a little more 00:29:01.950 --> 00:29:03.790 align:middle line:90% to the stock of capital. 00:29:03.790 --> 00:29:04.900 align:middle line:90% That's the way you grow. 00:29:04.900 --> 00:29:09.960 align:middle line:84% It's not that, all of a sudden, your stock of capital jumps. 00:29:09.960 --> 00:29:13.342 align:middle line:84% That's very much because this is a closed economy. 00:29:13.342 --> 00:29:15.300 align:middle line:84% If you're in an open economy, the capital stock 00:29:15.300 --> 00:29:17.370 align:middle line:84% can move a lot faster in a transition 00:29:17.370 --> 00:29:19.620 align:middle line:84% because you can borrow from abroad. 00:29:19.620 --> 00:29:23.280 align:middle line:84% You don't need to fund it all with domestic sources. 00:29:23.280 --> 00:29:25.830 align:middle line:84% And in fact, that's what typically happens 00:29:25.830 --> 00:29:29.460 align:middle line:84% in emerging markets and so on is they typically borrow 00:29:29.460 --> 00:29:30.880 align:middle line:90% for a long time. 00:29:30.880 --> 00:29:33.390 align:middle line:84% Problem is that they tend to consume it rather than invest 00:29:33.390 --> 00:29:35.723 align:middle line:84% it, and that's the reason you end up in financial crisis 00:29:35.723 --> 00:29:36.360 align:middle line:90% and so on. 00:29:36.360 --> 00:29:38.520 align:middle line:84% But in principle, things could go 00:29:38.520 --> 00:29:40.350 align:middle line:84% much faster if you have an open economy, 00:29:40.350 --> 00:29:43.040 align:middle line:84% and you have capital inflows into your country. 00:29:43.040 --> 00:29:48.320 align:middle line:84% But we'll talk more about that five or six lectures from now. 00:29:48.320 --> 00:29:50.570 align:middle line:84% But anyways, but this is what happens with an increase 00:29:50.570 --> 00:29:51.403 align:middle line:90% in the savings rate. 00:29:51.403 --> 00:29:54.830 align:middle line:84% So yes, it affects the rate of growth of the economy 00:29:54.830 --> 00:29:58.287 align:middle line:84% during the transition but not in the long run. 00:29:58.287 --> 00:29:59.870 align:middle line:84% Now, this transition can be very long. 00:29:59.870 --> 00:30:02.490 align:middle line:90% 00:30:02.490 --> 00:30:04.380 align:middle line:90% Now, what about consumption? 00:30:04.380 --> 00:30:11.610 align:middle line:84% So invariably-- and there's no way around that-- 00:30:11.610 --> 00:30:15.730 align:middle line:84% given a technology and so on, if the saving rate goes up, 00:30:15.730 --> 00:30:20.213 align:middle line:84% then output per worker will go up. 00:30:20.213 --> 00:30:21.630 align:middle line:84% The next question is, what happens 00:30:21.630 --> 00:30:22.890 align:middle line:90% to consumption per worker? 00:30:22.890 --> 00:30:27.460 align:middle line:84% Does consumption per worker go up or not? 00:30:27.460 --> 00:30:29.590 align:middle line:84% You are inclined to say, well, I mean, 00:30:29.590 --> 00:30:34.180 align:middle line:84% it makes sense that it goes up because we have more income. 00:30:34.180 --> 00:30:35.980 align:middle line:84% If the saving rate is little less times y, 00:30:35.980 --> 00:30:40.150 align:middle line:84% then consumption is 1 minus little s times Y. 00:30:40.150 --> 00:30:43.570 align:middle line:84% So income goes up, consumption should go up. 00:30:43.570 --> 00:30:48.840 align:middle line:90% 00:30:48.840 --> 00:30:52.400 align:middle line:84% And yes, that's a dominant source. 00:30:52.400 --> 00:30:55.250 align:middle line:84% But it's not all the story because remember 00:30:55.250 --> 00:30:56.180 align:middle line:90% what I told you. 00:30:56.180 --> 00:31:02.900 align:middle line:90% 00:31:02.900 --> 00:31:07.040 align:middle line:84% So consumption here is going to be equal to 1 minus little 00:31:07.040 --> 00:31:15.110 align:middle line:84% s times Y. So consumption per person will be that. 00:31:15.110 --> 00:31:18.540 align:middle line:84% Remember that what is increasing Y over N there-- 00:31:18.540 --> 00:31:20.700 align:middle line:84% so what is making this guy go up, 00:31:20.700 --> 00:31:24.170 align:middle line:84% which will lead to an increase in consumption over N-- 00:31:24.170 --> 00:31:28.650 align:middle line:90% is that this guy went up. 00:31:28.650 --> 00:31:30.810 align:middle line:84% And that's a force in the opposite direction. 00:31:30.810 --> 00:31:34.180 align:middle line:90% 00:31:34.180 --> 00:31:39.580 align:middle line:84% And in fact, that was one of the debates with the Southeast 00:31:39.580 --> 00:31:43.150 align:middle line:84% Asian miracle is that it was fueled by lots of savings. 00:31:43.150 --> 00:31:44.650 align:middle line:84% So people say, OK, that's wonderful. 00:31:44.650 --> 00:31:47.170 align:middle line:84% Your output growth is very fast, but consumption growth 00:31:47.170 --> 00:31:48.280 align:middle line:90% is not so fast. 00:31:48.280 --> 00:31:50.170 align:middle line:84% And at some point, it may be hurting you. 00:31:50.170 --> 00:31:52.503 align:middle line:84% I think that they were right, though, for other reasons. 00:31:52.503 --> 00:31:55.030 align:middle line:90% 00:31:55.030 --> 00:31:58.450 align:middle line:84% But that picture makes the point. 00:31:58.450 --> 00:32:02.320 align:middle line:84% So if your saving rate to start with-- this is a general lesson. 00:32:02.320 --> 00:32:06.880 align:middle line:84% If the saving rate you start with is very, very low, 00:32:06.880 --> 00:32:08.695 align:middle line:84% then an increase in the saving rate 00:32:08.695 --> 00:32:10.570 align:middle line:84% will lead to a strong increase in consumption 00:32:10.570 --> 00:32:13.630 align:middle line:84% because this change is small relative to the big bang 00:32:13.630 --> 00:32:16.900 align:middle line:84% get in output because, if you have low saving rate, 00:32:16.900 --> 00:32:21.710 align:middle line:84% that also means that the capital stock is very low. 00:32:21.710 --> 00:32:26.027 align:middle line:84% And if the capital stock is very low, f prime is very big. 00:32:26.027 --> 00:32:27.610 align:middle line:84% This is a concave function, and you're 00:32:27.610 --> 00:32:29.740 align:middle line:84% in the steep part of the function. 00:32:29.740 --> 00:32:32.380 align:middle line:84% Later on, if saving is very high, 00:32:32.380 --> 00:32:34.820 align:middle line:84% you're going to tend to have capital stock very high. 00:32:34.820 --> 00:32:37.870 align:middle line:84% And then, first of all, more capital 00:32:37.870 --> 00:32:40.360 align:middle line:84% won't increase output per worker a lot 00:32:40.360 --> 00:32:43.220 align:middle line:90% because of decreasing returns. 00:32:43.220 --> 00:32:46.090 align:middle line:84% And this is a big number, so it starts dominating. 00:32:46.090 --> 00:32:47.380 align:middle line:90% And that's what you see here. 00:32:47.380 --> 00:32:51.250 align:middle line:84% This economy has increased the saving rate. 00:32:51.250 --> 00:32:52.810 align:middle line:90% Consumption per worker rises. 00:32:52.810 --> 00:32:55.150 align:middle line:84% But at some point, it reaches a maximum, 00:32:55.150 --> 00:32:56.740 align:middle line:90% and then it starts declining. 00:32:56.740 --> 00:32:58.400 align:middle line:90% I mean, think of the limit. 00:32:58.400 --> 00:33:02.750 align:middle line:84% If you save 100% of your income, you don't consume anything. 00:33:02.750 --> 00:33:04.650 align:middle line:84% No matter how much is your output, 00:33:04.650 --> 00:33:07.310 align:middle line:84% if your saving rate is 100%, then 00:33:07.310 --> 00:33:09.870 align:middle line:84% you're not going to consume anything. 00:33:09.870 --> 00:33:14.210 align:middle line:84% If you have no income, no savings rate, no savings, 00:33:14.210 --> 00:33:15.980 align:middle line:84% no capital stock, no income, you're 00:33:15.980 --> 00:33:18.530 align:middle line:84% not going to consume anything either. 00:33:18.530 --> 00:33:20.988 align:middle line:84% So at least you know these two points. 00:33:20.988 --> 00:33:23.030 align:middle line:84% And since you know there are some positive points 00:33:23.030 --> 00:33:24.243 align:middle line:90% in the middle. 00:33:24.243 --> 00:33:25.910 align:middle line:84% You know that the curve is going to tend 00:33:25.910 --> 00:33:28.010 align:middle line:90% to have that kind of change. 00:33:28.010 --> 00:33:34.160 align:middle line:84% It's going to be nonmonotonic, and that's the way it is. 00:33:34.160 --> 00:33:37.610 align:middle line:84% So let me just play with a few numbers. 00:33:37.610 --> 00:33:41.640 align:middle line:84% This is-- yeah, let me play with a few numbers. 00:33:41.640 --> 00:33:44.110 align:middle line:90% It's not that crazy. 00:33:44.110 --> 00:33:47.470 align:middle line:84% Suppose you have a production function that gives equal weight 00:33:47.470 --> 00:33:48.740 align:middle line:90% to capital and workers-- 00:33:48.740 --> 00:33:51.690 align:middle line:90% so this production function. 00:33:51.690 --> 00:33:55.840 align:middle line:84% Does the production function have constant return to scale? 00:33:55.840 --> 00:33:58.820 align:middle line:84% It better be because that's what we're doing. 00:33:58.820 --> 00:34:01.640 align:middle line:90% What do you think? 00:34:01.640 --> 00:34:02.960 align:middle line:90% Yes. 00:34:02.960 --> 00:34:07.810 align:middle line:84% The sum of the exponents is 1, so it's K to the 1/2, 00:34:07.810 --> 00:34:08.889 align:middle line:90% N to the 1/2. 00:34:08.889 --> 00:34:10.929 align:middle line:84% The sum of the exponents is 1, so you 00:34:10.929 --> 00:34:14.389 align:middle line:84% know that it's proportional to a scaling factor. 00:34:14.389 --> 00:34:18.330 align:middle line:84% So we're going to use as a scaling-- 00:34:18.330 --> 00:34:24.785 align:middle line:90% as before, N. S we have this. 00:34:24.785 --> 00:34:28.760 align:middle line:84% This is a f of little f of K over N 00:34:28.760 --> 00:34:33.642 align:middle line:84% is the square root of K over N minus delta K over N. 00:34:33.642 --> 00:34:35.809 align:middle line:84% All that I'm doing is I'm plugging in that function. 00:34:35.809 --> 00:34:39.330 align:middle line:90% 00:34:39.330 --> 00:34:42.929 align:middle line:84% So here, I'm replacing all these functions 00:34:42.929 --> 00:34:46.679 align:middle line:84% by a specific example, one in which 00:34:46.679 --> 00:34:49.980 align:middle line:84% this is a square root of K over N. 00:34:49.980 --> 00:34:52.409 align:middle line:84% That's a concave function-- square root. 00:34:52.409 --> 00:34:56.340 align:middle line:90% 00:34:56.340 --> 00:34:57.900 align:middle line:90% Good. 00:34:57.900 --> 00:34:59.580 align:middle line:90% Now, do it as an exercise. 00:34:59.580 --> 00:35:01.200 align:middle line:84% If you solve for the steady state, 00:35:01.200 --> 00:35:02.783 align:middle line:84% how do you solve for the steady state? 00:35:02.783 --> 00:35:04.560 align:middle line:90% Well, set this equal to 0. 00:35:04.560 --> 00:35:08.060 align:middle line:84% That will give you the steady state. 00:35:08.060 --> 00:35:10.560 align:middle line:84% The steady state is when the capital is not growing anymore. 00:35:10.560 --> 00:35:12.870 align:middle line:90% It's when this is equal to 0. 00:35:12.870 --> 00:35:16.230 align:middle line:84% When this is equal to 0, I can solve for the steady state 00:35:16.230 --> 00:35:18.900 align:middle line:90% level of K over N from here. 00:35:18.900 --> 00:35:21.720 align:middle line:84% This equal to 0, I can solve for K over N. 00:35:21.720 --> 00:35:24.990 align:middle line:84% And I'm going to call that the steady state, K star. 00:35:24.990 --> 00:35:29.750 align:middle line:84% We typically use stars for steady states in growth theory. 00:35:29.750 --> 00:35:33.890 align:middle line:90% Well, the answer to this is K-- 00:35:33.890 --> 00:35:36.890 align:middle line:84% the steady state, stock of capital per person 00:35:36.890 --> 00:35:39.960 align:middle line:84% is the saving rate over delta squared. 00:35:39.960 --> 00:35:41.910 align:middle line:90% That's what it is. 00:35:41.910 --> 00:35:46.740 align:middle line:84% Output per person, which is the square root of K over N, 00:35:46.740 --> 00:35:50.230 align:middle line:84% is, therefore, the square root of s over delta squared, 00:35:50.230 --> 00:35:53.420 align:middle line:90% so it's over s delta. 00:35:53.420 --> 00:35:57.560 align:middle line:84% So in this particular model, in the long run, output per worker 00:35:57.560 --> 00:36:00.200 align:middle line:84% doubles when the savings rate doubles. 00:36:00.200 --> 00:36:03.435 align:middle line:84% If I double the saving rate, then output per worker 00:36:03.435 --> 00:36:03.935 align:middle line:90% will double. 00:36:03.935 --> 00:36:07.710 align:middle line:90% 00:36:07.710 --> 00:36:09.930 align:middle line:84% Notice that the stock of capital is 00:36:09.930 --> 00:36:15.380 align:middle line:84% going to grow a lot more when you increase the savings rate. 00:36:15.380 --> 00:36:16.020 align:middle line:90% It's squared. 00:36:16.020 --> 00:36:19.330 align:middle line:90% 00:36:19.330 --> 00:36:23.290 align:middle line:84% So in that economy, if you do increase the savings rate 00:36:23.290 --> 00:36:28.150 align:middle line:84% from 10% to 20%, this is the way it goes. 00:36:28.150 --> 00:36:32.800 align:middle line:84% So remember, 10% to 20%-- that means that the new steady state 00:36:32.800 --> 00:36:34.960 align:middle line:84% output per worker will be twice what it was 00:36:34.960 --> 00:36:37.420 align:middle line:90% in the previous steady state. 00:36:37.420 --> 00:36:39.710 align:middle line:90% So you go from 1 to 2. 00:36:39.710 --> 00:36:42.593 align:middle line:90% But it takes a long time. 00:36:42.593 --> 00:36:43.760 align:middle line:90% The numbers are [INAUDIBLE]. 00:36:43.760 --> 00:36:48.750 align:middle line:84% 50 years it takes you to go to the new state. 00:36:48.750 --> 00:36:52.138 align:middle line:84% And so that's the time frame we're talking about. 00:36:52.138 --> 00:36:53.930 align:middle line:84% So it is true that the saving rate will not 00:36:53.930 --> 00:37:00.290 align:middle line:84% change the long-run rate of growth, absent other mechanisms. 00:37:00.290 --> 00:37:03.740 align:middle line:84% But you can grow faster than your average, your steady state 00:37:03.740 --> 00:37:06.110 align:middle line:90% level for quite some time. 00:37:06.110 --> 00:37:13.880 align:middle line:84% And again, a lot of the Asian miracle has been of that kind. 00:37:13.880 --> 00:37:16.300 align:middle line:84% This is what I was telling you of China before, no? 00:37:16.300 --> 00:37:18.500 align:middle line:84% Well, yeah, you can grow very fast, 00:37:18.500 --> 00:37:21.250 align:middle line:84% especially if you have saving rate much higher than 20%-- 00:37:21.250 --> 00:37:23.920 align:middle line:90% I mean 50% or so. 00:37:23.920 --> 00:37:26.590 align:middle line:84% But the rate of growth will have a tendency 00:37:26.590 --> 00:37:28.325 align:middle line:84% to decline, absent some other miracle. 00:37:28.325 --> 00:37:30.700 align:middle line:84% There are a lot of the reasons why we have all this fight 00:37:30.700 --> 00:37:33.100 align:middle line:90% about technology and so on. 00:37:33.100 --> 00:37:35.860 align:middle line:84% It has to do with-- because that's the main mechanism-- 00:37:35.860 --> 00:37:40.020 align:middle line:84% alternative mechanism to grow is technology. 00:37:40.020 --> 00:37:42.670 align:middle line:84% We're going to talk about that in the next lecture. 00:37:42.670 --> 00:37:45.250 align:middle line:84% But this force, which is, what I said before, 00:37:45.250 --> 00:37:46.780 align:middle line:90% is the easy part of growth-- 00:37:46.780 --> 00:37:48.895 align:middle line:84% it's very difficult to fight this pattern. 00:37:48.895 --> 00:37:53.440 align:middle line:90% 00:37:53.440 --> 00:37:57.550 align:middle line:84% So here you have numbers for the steady states. 00:37:57.550 --> 00:38:01.050 align:middle line:84% So if the saving rate is 0, obviously, everything is 0-- 00:38:01.050 --> 00:38:03.090 align:middle line:90% no way around. 00:38:03.090 --> 00:38:07.260 align:middle line:84% If saving rate is 0.1, 10%, then, in this model, 00:38:07.260 --> 00:38:08.670 align:middle line:90% capital per worker is 1. 00:38:08.670 --> 00:38:10.470 align:middle line:90% Output per worker is 1. 00:38:10.470 --> 00:38:12.670 align:middle line:84% Consumption per worker then go from 0 to 1. 00:38:12.670 --> 00:38:13.170 align:middle line:90% Why? 00:38:13.170 --> 00:38:14.587 align:middle line:84% Because you were saving something. 00:38:14.587 --> 00:38:17.890 align:middle line:84% So it's 1 minus 0.1, which is a saving rate. 00:38:17.890 --> 00:38:20.320 align:middle line:84% Suppose you double the saving rate. 00:38:20.320 --> 00:38:23.063 align:middle line:84% Well, we know that we're going to double output per worker 00:38:23.063 --> 00:38:23.730 align:middle line:90% in this economy. 00:38:23.730 --> 00:38:24.272 align:middle line:90% We said that. 00:38:24.272 --> 00:38:25.830 align:middle line:90% We're going to go from 1 to 2. 00:38:25.830 --> 00:38:27.300 align:middle line:84% The capital stock is going to have 00:38:27.300 --> 00:38:31.290 align:middle line:84% to grow a lot more to double the amount of output. 00:38:31.290 --> 00:38:32.270 align:middle line:90% Why is that? 00:38:32.270 --> 00:38:33.860 align:middle line:90% Decreasing returns. 00:38:33.860 --> 00:38:36.740 align:middle line:84% To double output, you're going to have to much more than double 00:38:36.740 --> 00:38:41.810 align:middle line:84% capital because you're going to be fighting decreasing returns. 00:38:41.810 --> 00:38:44.060 align:middle line:90% What about consumption? 00:38:44.060 --> 00:38:45.620 align:middle line:84% Well, it won't double because you're 00:38:45.620 --> 00:38:47.870 align:middle line:84% doing this out of increasing the saving rate. 00:38:47.870 --> 00:38:52.200 align:middle line:84% So you get the 2 minus, now, 0.2, not 0.1-- 00:38:52.200 --> 00:38:57.780 align:middle line:84% minus 0.2 times 2, so you get 1.6 and so on. 00:38:57.780 --> 00:39:02.550 align:middle line:84% And the higher you go with your saving rate, 00:39:02.550 --> 00:39:09.150 align:middle line:84% the harder it gets for capital to bring along 00:39:09.150 --> 00:39:12.030 align:middle line:84% output per capita, and the more the drag 00:39:12.030 --> 00:39:15.030 align:middle line:84% on consumption because you need to be saving a lot in order 00:39:15.030 --> 00:39:18.510 align:middle line:84% to maintain this high stock of capital that you're having. 00:39:18.510 --> 00:39:20.640 align:middle line:84% You have a very large stock of capital. 00:39:20.640 --> 00:39:23.010 align:middle line:84% That means you need to save a lot just 00:39:23.010 --> 00:39:26.010 align:middle line:84% for the sake of maintaining that stock of capital. 00:39:26.010 --> 00:39:32.610 align:middle line:84% And so little is left for extra output per capita. 00:39:32.610 --> 00:39:36.420 align:middle line:84% And so you see that, here, for this particular model, 00:39:36.420 --> 00:39:39.120 align:middle line:84% when the saving rate exceeds 0.5, 00:39:39.120 --> 00:39:42.540 align:middle line:84% then output obviously keeps rising 00:39:42.540 --> 00:39:44.400 align:middle line:84% when you increase the saving rate, 00:39:44.400 --> 00:39:46.090 align:middle line:90% but output starts declining. 00:39:46.090 --> 00:39:48.520 align:middle line:90% So you're in the declining part. 00:39:48.520 --> 00:39:51.580 align:middle line:84% And if you get to 1, of course, there is no consumption. 00:39:51.580 --> 00:39:53.720 align:middle line:84% So that's the curve that we trace. 00:39:53.720 --> 00:40:03.590 align:middle line:90% 00:40:03.590 --> 00:40:04.430 align:middle line:90% Is everything clear? 00:40:04.430 --> 00:40:06.920 align:middle line:84% Now I'm going to-- that's the basic Solow model. 00:40:06.920 --> 00:40:12.810 align:middle line:84% And that's a model that, again, you need to control completely. 00:40:12.810 --> 00:40:15.190 align:middle line:84% All that I'm going to do now is very simple. 00:40:15.190 --> 00:40:20.430 align:middle line:84% I'm going to just modify a little bit this model to add 00:40:20.430 --> 00:40:22.890 align:middle line:90% population growth. 00:40:22.890 --> 00:40:27.780 align:middle line:84% So what happens-- by the way, for centuries, 00:40:27.780 --> 00:40:31.770 align:middle line:84% population growth has been one of the main-- in this model, 00:40:31.770 --> 00:40:37.492 align:middle line:84% we concluded that output per worker was not growing. 00:40:37.492 --> 00:40:39.940 align:middle line:84% What we're going to conclude in a second 00:40:39.940 --> 00:40:43.020 align:middle line:84% is that output per worker will not 00:40:43.020 --> 00:40:45.580 align:middle line:90% grow if population is growing. 00:40:45.580 --> 00:40:48.470 align:middle line:84% But that means that output is growing. 00:40:48.470 --> 00:40:50.700 align:middle line:84% If population is growing, and output per worker 00:40:50.700 --> 00:40:54.060 align:middle line:84% is not growing, it's constant, that means output is also 00:40:54.060 --> 00:40:54.730 align:middle line:90% growing. 00:40:54.730 --> 00:40:59.250 align:middle line:84% And for a long time, growth of output, 00:40:59.250 --> 00:41:01.710 align:middle line:84% not of output per worker, was driven 00:41:01.710 --> 00:41:04.020 align:middle line:90% by large population growth. 00:41:04.020 --> 00:41:06.180 align:middle line:84% And sometimes, you get big migration flows 00:41:06.180 --> 00:41:09.040 align:middle line:84% into a country that leads to growth and so on. 00:41:09.040 --> 00:41:13.450 align:middle line:84% Now big parts of the world have negative population growth. 00:41:13.450 --> 00:41:15.990 align:middle line:84% So now we're going through a cycle in which things 00:41:15.990 --> 00:41:20.100 align:middle line:84% are going the other way around in many large parts 00:41:20.100 --> 00:41:20.820 align:middle line:90% of the world. 00:41:20.820 --> 00:41:24.840 align:middle line:84% This is true in almost all of continental Europe, 00:41:24.840 --> 00:41:26.070 align:middle line:90% certainly in Japan. 00:41:26.070 --> 00:41:30.480 align:middle line:84% I said South Korea, China, and even some places 00:41:30.480 --> 00:41:31.950 align:middle line:90% in Latin America. 00:41:31.950 --> 00:41:36.710 align:middle line:84% So the drag, actually, is against that. 00:41:36.710 --> 00:41:38.840 align:middle line:84% We don't have the natural force for growth 00:41:38.840 --> 00:41:42.370 align:middle line:84% that we had for many, many years. 00:41:42.370 --> 00:41:44.840 align:middle line:84% So let me introduce population growth. 00:41:44.840 --> 00:41:48.370 align:middle line:84% So assume now that population, rather than being constant, 00:41:48.370 --> 00:41:51.380 align:middle line:84% grows at the related gN, which could be positive or negative. 00:41:51.380 --> 00:41:55.330 align:middle line:84% I'm going to do the example for a positive population growth 00:41:55.330 --> 00:41:56.120 align:middle line:90% example. 00:41:56.120 --> 00:41:59.260 align:middle line:84% So there's no equation that changes, in the sense 00:41:59.260 --> 00:42:01.490 align:middle line:90% that this is still true. 00:42:01.490 --> 00:42:04.570 align:middle line:84% It's still true that investment is equal to savings. 00:42:04.570 --> 00:42:10.540 align:middle line:84% It's still true that output is equal to output per worker, 00:42:10.540 --> 00:42:16.210 align:middle line:84% and output is equal to f of K and N and so on and so forth. 00:42:16.210 --> 00:42:19.270 align:middle line:84% The thing that is a little trickier 00:42:19.270 --> 00:42:26.365 align:middle line:84% is that, in this model, if I don't normalize things for-- 00:42:26.365 --> 00:42:29.080 align:middle line:90% 00:42:29.080 --> 00:42:32.330 align:middle line:84% in this case here, where population was not growing, 00:42:32.330 --> 00:42:35.320 align:middle line:84% I could have just eliminated this N as a constant. 00:42:35.320 --> 00:42:37.480 align:middle line:84% And I would have done everything in capital, 00:42:37.480 --> 00:42:39.328 align:middle line:84% in a space of capital here, and output here 00:42:39.328 --> 00:42:40.870 align:middle line:84% would have been the same, just scaled 00:42:40.870 --> 00:42:43.970 align:middle line:90% by a number, a constant N. 00:42:43.970 --> 00:42:46.010 align:middle line:84% When I have population growth, I'm 00:42:46.010 --> 00:42:49.700 align:middle line:84% not indifferent between doing one way or the other because, 00:42:49.700 --> 00:42:50.810 align:middle line:90% if I don't have-- 00:42:50.810 --> 00:42:55.700 align:middle line:84% if I do it in the space of K and Y, and population is growing, 00:42:55.700 --> 00:42:58.265 align:middle line:84% then all these curves are moving. 00:42:58.265 --> 00:42:59.640 align:middle line:84% This is a very unfriendly diagram 00:42:59.640 --> 00:43:01.140 align:middle line:84% because my curves are all moving. 00:43:01.140 --> 00:43:03.490 align:middle line:84% As N is moving, everything is moving. 00:43:03.490 --> 00:43:05.198 align:middle line:84% So the trick in all these growth models, 00:43:05.198 --> 00:43:07.740 align:middle line:84% and it's going to be even more important in the next lecture, 00:43:07.740 --> 00:43:10.570 align:middle line:84% is to find the right scaling of capital 00:43:10.570 --> 00:43:13.740 align:middle line:84% so there is a steady state, so your curves 00:43:13.740 --> 00:43:16.670 align:middle line:84% are not moving around as population growth. 00:43:16.670 --> 00:43:18.445 align:middle line:84% It's very easy to find the scaling factor. 00:43:18.445 --> 00:43:21.010 align:middle line:90% It's population. 00:43:21.010 --> 00:43:25.320 align:middle line:90% So that's what I'm going to do. 00:43:25.320 --> 00:43:28.070 align:middle line:84% But remember what is different here is-- 00:43:28.070 --> 00:43:30.110 align:middle line:84% so what I'm saying is I want to get 00:43:30.110 --> 00:43:33.103 align:middle line:84% all my variables scaled by population 00:43:33.103 --> 00:43:34.020 align:middle line:90% at some point in time. 00:43:34.020 --> 00:43:36.067 align:middle line:84% That's what I want to do because I know-- 00:43:36.067 --> 00:43:38.150 align:middle line:84% I practice enough with these things-- that's going 00:43:38.150 --> 00:43:39.233 align:middle line:90% to give me a steady state. 00:43:39.233 --> 00:43:41.810 align:middle line:90% 00:43:41.810 --> 00:43:43.670 align:middle line:84% Now, what is tricky relative to what 00:43:43.670 --> 00:43:46.940 align:middle line:84% I showed you before is that, before, I just divided by N 00:43:46.940 --> 00:43:49.610 align:middle line:90% both sides, and I was home. 00:43:49.610 --> 00:43:51.440 align:middle line:90% Now I can't really do that. 00:43:51.440 --> 00:43:55.230 align:middle line:84% OK, let me divide by Nt plus 1 both sides. 00:43:55.230 --> 00:43:56.880 align:middle line:90% So that's nice. 00:43:56.880 --> 00:44:01.270 align:middle line:84% I get my capital per worker at t plus 1. 00:44:01.270 --> 00:44:03.558 align:middle line:84% But there are certain things that are not as nice. 00:44:03.558 --> 00:44:06.100 align:middle line:84% What I have on the right-hand side is not what I really want. 00:44:06.100 --> 00:44:11.150 align:middle line:84% I don't want capital over population next period. 00:44:11.150 --> 00:44:15.520 align:middle line:84% My steady state is going to be in the space of capital 00:44:15.520 --> 00:44:17.710 align:middle line:90% overpopulation at the same time. 00:44:17.710 --> 00:44:20.390 align:middle line:90% That's my steady state. 00:44:20.390 --> 00:44:22.940 align:middle line:90% So this is not so nice. 00:44:22.940 --> 00:44:24.880 align:middle line:84% So what I have to do is I want to convert 00:44:24.880 --> 00:44:26.890 align:middle line:84% this, the right-hand side, in assumption 00:44:26.890 --> 00:44:29.240 align:middle line:84% that these are the kind of things that I want to have. 00:44:29.240 --> 00:44:31.000 align:middle line:84% So what I'm going to do is divide 00:44:31.000 --> 00:44:35.560 align:middle line:84% and multiply each of these sides by Nt over Nt plus 1-- 00:44:35.560 --> 00:44:37.090 align:middle line:90% sorry-- I'm going to divide. 00:44:37.090 --> 00:44:41.020 align:middle line:90% I multiply each of these by Nt. 00:44:41.020 --> 00:44:47.120 align:middle line:84% So multiply by Nt, divide by Nt, so multiplying by 1. 00:44:47.120 --> 00:44:49.520 align:middle line:84% And then I can rearrange the terms in this way 00:44:49.520 --> 00:44:54.310 align:middle line:84% so I get what I want, which is capital per person at time t, 00:44:54.310 --> 00:44:55.060 align:middle line:90% all at time t. 00:44:55.060 --> 00:44:58.800 align:middle line:90% But then I get this ratio here. 00:44:58.800 --> 00:45:01.870 align:middle line:84% And I can do the same for this expression here. 00:45:01.870 --> 00:45:03.165 align:middle line:90% Now, what is that ratio? 00:45:03.165 --> 00:45:06.060 align:middle line:90% 00:45:06.060 --> 00:45:09.925 align:middle line:84% It's population today divided by population tomorrow. 00:45:09.925 --> 00:45:14.960 align:middle line:90% 00:45:14.960 --> 00:45:21.050 align:middle line:84% Well, it's 1 over 1 plus the rate of growth of population. 00:45:21.050 --> 00:45:25.310 align:middle line:84% Nt plus 1 is equal to Nt times 1 plus gN. 00:45:25.310 --> 00:45:27.680 align:middle line:84% That's the rate of growth of population. 00:45:27.680 --> 00:45:44.410 align:middle line:90% 00:45:44.410 --> 00:45:49.390 align:middle line:84% So what I have here is 1 over 1 plus gN. 00:45:49.390 --> 00:45:51.950 align:middle line:90% Now, gN is not a big number. 00:45:51.950 --> 00:46:01.120 align:middle line:84% So 1 over 1 plus gN is approximately equal to minus gN. 00:46:01.120 --> 00:46:04.150 align:middle line:84% So 1 over 1 plus gN-- gN is very close to 0-- 00:46:04.150 --> 00:46:08.110 align:middle line:84% is approximately equal to minus gN. 00:46:08.110 --> 00:46:12.850 align:middle line:84% So that's the reason this guy became that guy-- approximately 00:46:12.850 --> 00:46:14.320 align:middle line:90% that guy. 00:46:14.320 --> 00:46:16.600 align:middle line:84% I can do the same here, but it turns out 00:46:16.600 --> 00:46:21.010 align:middle line:84% that there's an extra term here which 00:46:21.010 --> 00:46:27.670 align:middle line:84% is equal to s times gN times Yt over Nt. 00:46:27.670 --> 00:46:28.870 align:middle line:90% Well, that's second-order. 00:46:28.870 --> 00:46:30.760 align:middle line:84% That's the reason I'm going to drop it. 00:46:30.760 --> 00:46:32.974 align:middle line:90% It's a saving rate which is-- 00:46:32.974 --> 00:46:38.080 align:middle line:84% it's a small number times a rate of population growth, which 00:46:38.080 --> 00:46:40.490 align:middle line:84% is a number like 0.01 or something like that. 00:46:40.490 --> 00:46:42.820 align:middle line:84% So that's a small number, so I'm dropping it. 00:46:42.820 --> 00:46:45.460 align:middle line:84% That's a bigger approximation than that one, actually, 00:46:45.460 --> 00:46:46.600 align:middle line:90% but I'm going to do it. 00:46:46.600 --> 00:46:48.440 align:middle line:84% Everything becomes a lot simpler. 00:46:48.440 --> 00:46:50.530 align:middle line:90% So this is an approximation. 00:46:50.530 --> 00:46:54.440 align:middle line:84% I'm just dropping second-order terms. 00:46:54.440 --> 00:46:56.320 align:middle line:84% And once I have that, I have the system 00:46:56.320 --> 00:46:58.930 align:middle line:84% I want because now I have a system 00:46:58.930 --> 00:47:10.230 align:middle line:84% for the evolution of the capital per worker or per person. 00:47:10.230 --> 00:47:12.870 align:middle line:84% And if you see, it looks exactly as we had before. 00:47:12.870 --> 00:47:16.190 align:middle line:84% Remember, this is exactly what we had before-- 00:47:16.190 --> 00:47:20.150 align:middle line:84% sf K over-- we used to have N subscript t. 00:47:20.150 --> 00:47:22.250 align:middle line:90% Now it's K over Nt. 00:47:22.250 --> 00:47:26.000 align:middle line:84% But what is different is that now, rather than having only 00:47:26.000 --> 00:47:28.790 align:middle line:84% the depreciation rate here, we have the depreciation rate 00:47:28.790 --> 00:47:32.820 align:middle line:84% plus the rate of growth of population. 00:47:32.820 --> 00:47:35.240 align:middle line:84% Why do you think we have the rate of growth of population 00:47:35.240 --> 00:47:36.600 align:middle line:90% there? 00:47:36.600 --> 00:47:42.060 align:middle line:84% Remember the economics behind this expression before. 00:47:42.060 --> 00:47:49.710 align:middle line:84% It was this is what adds to capital, to capital per worker. 00:47:49.710 --> 00:47:51.900 align:middle line:84% This is what you need to maintain, 00:47:51.900 --> 00:47:55.170 align:middle line:90% what takes away from capital. 00:47:55.170 --> 00:47:58.143 align:middle line:90% Now, it's what takes away from-- 00:47:58.143 --> 00:48:00.810 align:middle line:84% given we're doing everything in the space of capital per worker, 00:48:00.810 --> 00:48:02.380 align:middle line:90% that takes away from capital-- 00:48:02.380 --> 00:48:04.140 align:middle line:90% oh, that's a typo. 00:48:04.140 --> 00:48:04.965 align:middle line:90% There's a t there-- 00:48:04.965 --> 00:48:07.380 align:middle line:90% 00:48:07.380 --> 00:48:07.880 align:middle line:90% t. 00:48:07.880 --> 00:48:12.540 align:middle line:90% 00:48:12.540 --> 00:48:15.930 align:middle line:84% So why do you think I have this gN here? 00:48:15.930 --> 00:48:19.200 align:middle line:84% Well, I have only one minute, so I don't have time to-- 00:48:19.200 --> 00:48:24.900 align:middle line:84% because if I want to maintain a stock of capital per worker, 00:48:24.900 --> 00:48:29.980 align:middle line:84% and workers are growing, then I need to be growing the capital. 00:48:29.980 --> 00:48:31.740 align:middle line:84% So even if I had no depreciation, 00:48:31.740 --> 00:48:34.890 align:middle line:84% if I want to maintain the capital per worker constant, 00:48:34.890 --> 00:48:37.500 align:middle line:84% and workers are growing, then I need 00:48:37.500 --> 00:48:39.390 align:middle line:90% to grow the stock of capital. 00:48:39.390 --> 00:48:41.460 align:middle line:84% So in order to maintain the capital-- 00:48:41.460 --> 00:48:44.460 align:middle line:84% I still need to spend what I used to spend for depreciation 00:48:44.460 --> 00:48:46.020 align:middle line:90% of the capital stock. 00:48:46.020 --> 00:48:50.400 align:middle line:84% But if I want to maintain the capital per worker constant, 00:48:50.400 --> 00:48:55.350 align:middle line:84% then I'm going to need more investment, just to make up 00:48:55.350 --> 00:48:58.000 align:middle line:90% for that extra component. 00:48:58.000 --> 00:49:01.560 align:middle line:90% So now, set gA equal to 0. 00:49:01.560 --> 00:49:04.170 align:middle line:84% Your diagram is exactly as before in this space. 00:49:04.170 --> 00:49:06.400 align:middle line:90% Set a equal to 1 and constant. 00:49:06.400 --> 00:49:11.860 align:middle line:84% But this line, the red line here will have delta plus gN. 00:49:11.860 --> 00:49:13.360 align:middle line:90% So it rotates up. 00:49:13.360 --> 00:49:16.940 align:middle line:90% 00:49:16.940 --> 00:49:18.860 align:middle line:84% So you can play here and see what 00:49:18.860 --> 00:49:21.180 align:middle line:84% happens if there is a change in population growth 00:49:21.180 --> 00:49:23.058 align:middle line:90% and so on and so forth. 00:49:23.058 --> 00:49:24.850 align:middle line:84% It's going to be counterintuitive initially 00:49:24.850 --> 00:49:27.680 align:middle line:84% because you see, if I increase population growth, 00:49:27.680 --> 00:49:30.280 align:middle line:84% this curve will rotate up, and then 00:49:30.280 --> 00:49:32.890 align:middle line:84% it will appear as if that leads to negative growth. 00:49:32.890 --> 00:49:35.630 align:middle line:90% 00:49:35.630 --> 00:49:37.160 align:middle line:84% But you don't get negative growth. 00:49:37.160 --> 00:49:43.505 align:middle line:84% In this diagram, you do get that Y over N will decline. 00:49:43.505 --> 00:49:45.630 align:middle line:84% But that doesn't mean that you get negative growth. 00:49:45.630 --> 00:49:50.440 align:middle line:84% It just means that output is not growing as fast as population. 00:49:50.440 --> 00:49:53.170 align:middle line:84% But both are growing, just the population is growing faster 00:49:53.170 --> 00:49:53.920 align:middle line:90% than output. 00:49:53.920 --> 00:49:57.180 align:middle line:90% I'll start from that-- 00:49:57.180 --> 00:49:58.750 align:middle line:84% oh, I think it's after your break, 00:49:58.750 --> 00:50:01.600 align:middle line:84% so when you have forgotten everything by then. 00:50:01.600 --> 00:50:04.750 align:middle line:84% So I'll do a review of this, and then we-- 00:50:04.750 --> 00:50:07.590 align:middle line:90% OK, have a nice break. 00:50:07.590 --> 00:50:20.000 align:middle line:90%