1 00:00:00,000 --> 00:00:02,800 Hi, we are now ready for our first real growth model. 2 00:00:02,800 --> 00:00:07,200 This is actually going to be a simple, [laugh] it's a Simple Growth Model. 3 00:00:07,200 --> 00:00:12,423 But it's still going to be far more complicated than any model we have seen so far.It's got all sorts of moving parts. 4 00:00:12,423 --> 00:00:18,962 That said, still by economists standards, by the models that economists use today, it is still relatively simple. 5 00:00:19,085 --> 00:00:20,415 Alright, so hang in there! 6 00:00:20,646 --> 00:00:21,515 So, here's how it's going to work. 7 00:00:21,592 --> 00:00:24,231 Model's going to have a group of Workers and there's going be these Coconut trees. 8 00:00:24,231 --> 00:00:26,946 And the Workers could pick the Coconuts 9 00:00:26,946 --> 00:00:29,877 When they pick the Coconuts, they can do two things with them. 10 00:00:29,877 --> 00:00:32,177 They can eat them, because they are so good and drink them up. 11 00:00:32,177 --> 00:00:33,500 So that they can live off these Coconuts. 12 00:00:33,500 --> 00:00:37,900 And they can also, instead of eating the Coconuts, use these Coconuts to build Machines. 13 00:00:37,900 --> 00:00:42,200 Right, and these Machines can help them pick Coconuts even faster. 14 00:00:42,200 --> 00:00:45,431 Now there's one other assumption that I'm going to add in. Right. 15 00:00:45,431 --> 00:00:48,331 The machines will wear out over time, because they are made out of coconuts, right? 16 00:00:48,331 --> 00:00:51,200 So, they sort of degrade over time and will need to be replaced with new machines. 17 00:00:51,200 --> 00:00:56,900 So, that's it - very simple economy: Coconuts, Workers, Machines, Machines depreciate. 18 00:00:56,900 --> 00:00:59,923 What we want to do is construct a Model of that. 19 00:00:59,923 --> 00:01:04,100 And we're going to use that Model to explain the role that investment in capital. 20 00:01:04,100 --> 00:01:08,200 In this case the Machines - produces growth, and the limits of that. 21 00:01:08,200 --> 00:01:11,100 Then from that we are going to see why innovation is so important. 22 00:01:11,100 --> 00:01:13,492 For now, we just want to focus on this very simple Model. 23 00:01:13,492 --> 00:01:16,046 Let's get started. Lots of moving parts. 24 00:01:16,046 --> 00:01:18,592 Tons of them. [laughter] So hang in there! 25 00:01:18,592 --> 00:01:22,731 There's going to be Workers time t , L of t for Laborers. 26 00:01:22,731 --> 00:01:27,323 Then there is going to be Machines at time t. We'll call that M of t for Machines ("Mt" = Machines at time t) 27 00:01:27,323 --> 00:01:32,808 The reason we put those 't's down there, those subscript 't' , is because over time there is change. 28 00:01:32,808 --> 00:01:35,100 Workers at Time 1, Workers at Time 2 29 00:01:35,100 --> 00:01:37,277 Machines at Time 1, Machines at Time 2, and so on. 30 00:01:37,277 --> 00:01:40,000 The Workers of Machine are going to combine to form Output. 31 00:01:40,000 --> 00:01:43,054 We are going to call that O of time t. (Ot = Output of Coconuts at time t) 32 00:01:43,162 --> 00:01:48,515 Then there's going to Coconuts that can be eaten, that's E of t (Et = Number consumed at time t) 33 00:01:48,623 --> 00:01:53,085 Or they can be Invested. That' s I of t, and when I say Invested, they can be turned into more Machines (It = Number invested at time t) 34 00:01:53,269 --> 00:01:56,838 Now, to figure out how many can be eaten and invested, there's going to be a Savings Rate (s = savings rate) 35 00:01:56,946 --> 00:02:01,500 The Savings Rate will determine the percentage of savings which we will go into Investment, 36 00:02:01,500 --> 00:02:04,427 and the amount we don't save that we will just eat, and that's "Et". 37 00:02:04,519 --> 00:02:07,292 The last thing in the model, I realize this is a lot [laughter], 38 00:02:07,338 --> 00:02:10,562 is the Depreciation rate (d = Depreciation rate), and that is the rate at which these machines wear out. 39 00:02:10,562 --> 00:02:14,377 We are going to assume there is some fixed rate percentage of the machines wear out each period. 40 00:02:14,515 --> 00:02:17,512 It's a simplification, but we are going to use it, All right. 41 00:02:17,635 --> 00:02:24,631 So, lot's of stuff. Workers, Machines, Coconuts. The Coconuts get eaten or turned into more machines, and there's Depreciation (percentage of the machines wear out each period). 42 00:02:24,985 --> 00:02:27,462 Got to make some assumptions. 43 00:02:27,538 --> 00:02:33,669 The first assumption we're going to make is that the production of these Coconuts is increasing, but Concave (Ot=(√Lt)(√Mt)) . 44 00:02:33,792 --> 00:02:37,738 Remember we have Concave Functions going up, but falling off - in both workers and and in machines. 45 00:02:37,754 --> 00:02:41,631 That means - the more machines, the more coconuts. The more workers, the more coconuts. 46 00:02:41,631 --> 00:02:43,062 But those things sort of fall off. 47 00:02:43,062 --> 00:02:45,308 Right. We're going to use a specific functional form that says: 48 00:02:45,308 --> 00:02:50,931 'The square root of the Laborers times the square root of the Number of Machines.' ((√Laborers)(√Machines)) 49 00:02:51,177 --> 00:02:57,015 Second Assumption: Output is either Consumed or turned into Machines. 50 00:02:57,108 --> 00:03:00,562 The Coconuts are either eaten or turned into Coconut Picking Machines. There is no waste. 51 00:03:00,654 --> 00:03:06,400 So Basically that means Output O is just equal to E plus I (Eaten + Invested). ("Ot"="Et"+"It"). 52 00:03:06,400 --> 00:03:15,738 Another way to write this is (I=s(O)). 53 00:03:15,738 --> 00:03:22,094 Because S is our Savings Rate and O is our Total Output . So, the amount Invest is going to be equal to our (Savings Rate)(Total Output). 54 00:03:22,171 --> 00:03:25,623 Assumption 3: The last thing is that these Machines Depreciate. Right. 55 00:03:25,777 --> 00:03:35,204 So, the Machines we have at "Mt" plus 1 are going to be the Machines we have at "Mt" plus Our Investment minus how many ever Depreciate. (Mt+1 = Mt + It - d(Mt)). 56 00:03:35,204 --> 00:03:41,031 Lot's going on. We have all these variables, and these are the equations that help us make sense of the variables. 57 00:03:41,031 --> 00:03:44,400 Let's step back to tha First Assumption - Concave. 58 00:03:44,400 --> 00:03:48,915 So, Concave means that like the first workers, you know gives you more Coconuts. 59 00:03:48,915 --> 00:03:54,200 The second give you more coconuts, but he gives you fewer Coconuts than the first one did. It sort of 'falls off'. 60 00:03:54,200 --> 00:03:58,000 Economists call this Diminishing Returns to Scale. 61 00:03:58,000 --> 00:04:04,115 Here's a picture - This is the number of Workers, and this is the number of Coconuts 62 00:04:04,300 --> 00:04:12,392 What we can see is this first worker gives you quite a bit. The second gives you fewer. The thirds gives you sort of even less. 63 00:04:12,485 --> 00:04:14,400 So what you get, is you add more workers. 64 00:04:14,400 --> 00:04:18,200 Yes you get more coconuts, but the workers become less and less valuable. 65 00:04:18,200 --> 00:04:20,500 The same is going to be true of machines. 66 00:04:20,500 --> 00:04:22,254 That's all Concave means. 67 00:04:22,362 --> 00:04:25,311 Okay. Alot going on here. Let's simplify things a bit. 68 00:04:25,404 --> 00:04:28,683 Let's assume that we just got a hundred workers. 69 00:04:28,821 --> 00:04:34,100 So, remember before Output was equal to the ((√Lt) (√Mt)). 70 00:04:34,100 --> 00:04:38,800 Were going to assume we have a hundred workers. That's going to mean it's going to be (10√Mt) . That will make things simpler. 71 00:04:38,800 --> 00:04:45,800 In a more realistic model. We would have workers deciding to goto work depending on what the wage is. 72 00:04:45,800 --> 00:04:52,200 So we would have to create a market for wages as a function of output, as a function of how much they like the Coconuts, and that would get really complicated. 73 00:04:52,200 --> 00:04:54,331 [laughter] So, we are just going to skip all that stuff. 74 00:04:54,331 --> 00:05:00,062 We're going to skip the entire labor market, and just assume everybody goes to work everyday, and then see what happens. 75 00:05:00,154 --> 00:05:05,500 Let's do an example. Now, were going to do some math. We'll try and do it slowly. 76 00:05:05,500 --> 00:05:13,000 So what we're going to do is assume the Depreciation rate is one forth (0.25), and the Savings rate is 30% (d=0.25, s=0.3), (Year 1: 4 machines). 77 00:05:13,000 --> 00:05:20,600 Let's suppose we start out with four machines. The the output is ten times the square root of four ((10)(√4)=20). 78 00:05:20,600 --> 00:05:23,100 So how much do they invest? 79 00:05:23,100 --> 00:05:30,392 Well, remember they invest 20 times 30%, so they invest in 6 machines. So, Investment = 6 ((20 30%)=6). 80 00:05:30,438 --> 00:05:33,223 How much Depreciation is there? 81 00:05:33,254 --> 00:05:36,000 Well, Depreciation is on the old machines. 82 00:05:36,000 --> 00:05:43,400 There were four machines in the past. 25% or one forth of those get worn out. That represents 1 machine. 83 00:05:43,400 --> 00:05:49,500 We subtract those two, and get a net of +5 machines ((20(0.3)) - (4(0.25)) = 5 or (Investment) - (Depreciation) = +5 machines. 84 00:05:49,500 --> 00:05:55,000 We're going to invest in 6 new ones. We loose one to Depreciation, so that gives us 5. 85 00:05:55,000 --> 00:06:00,100 We started out with 4 machines. So that means in Year 2 we have 9 machines . 86 00:06:00,100 --> 00:06:01,862 Here's how our economy works. 87 00:06:01,862 --> 00:06:04,077 We've got four machines, we produce twenty coconuts. 88 00:06:04,077 --> 00:06:06,500 We ate fourteen. We invested six. 89 00:06:06,500 --> 00:06:09,100 We lost one machine to depreciation. 90 00:06:09,192 --> 00:06:13,400 So now we have five new machines (6-1) 91 00:06:13,400 --> 00:06:18,500 That gives us a total of nine (4+(6-1)). We started out with four machines, now we got nine. 92 00:06:18,500 --> 00:06:23,946 We have a nice GDP of 20, and now we've added more machines so we should do better. 93 00:06:23,977 --> 00:06:25,815 So, let's look at the next year. 94 00:06:25,815 --> 00:06:29,854 The next year (Year 3) we've got nine machines. Output is ten times the square root of nine. 95 00:06:30,100 --> 00:06:32,123 So that's going to be thirty ((10)(√9) = 30). 96 00:06:32,277 --> 00:06:35,377 And let's think about how many new machines do we get. 97 00:06:35,377 --> 00:06:39,200 Well, we're (30)(0.3), that is how much we are going to save. 98 00:06:39,200 --> 00:06:43,800 So that gives us 9 new machines we're going to buy 99 00:06:43,800 --> 00:06:47,500 But the questions is 'how many do we loose to depreciation'? 100 00:06:47,500 --> 00:06:52,431 We had 9 machines and we multiply that times 0.25. 101 00:06:52,431 --> 00:06:56,400 That's like nine over four (9/4 = 2.25), so that's like 2 1/4. 102 00:06:56,400 --> 00:06:59,323 Let's just simplify this, and suppose it's 2. (fudge factor) 103 00:06:59,323 --> 00:07:12,985 The Depreciation is 2. So we had 9 machines to start. We get 7 new ones (9-2=7) for a total of 16 machines. With that little fudge of 1/4 to keep the math simpler. 104 00:07:12,985 --> 00:07:15,200 Let's take a look at our GDP. 105 00:07:15,200 --> 00:07:17,000 The previous period, our GDP was 20. 106 00:07:17,000 --> 00:07:19,100 Now our GDP is 30. 107 00:07:19,100 --> 00:07:21,000 So, we have this nice sustained growth. 108 00:07:21,000 --> 00:07:26,600 Let's look at Year 3. Year three began with 16 machines. 109 00:07:26,600 --> 00:07:32,500 If we have 16 machines, that means our Total Output equals 100 time square root of 16 which is 40 (Output=100√16=40) 110 00:07:32,500 --> 00:07:35,400 So, what's happened to our growth? We started out with 20, then 30, then 40. 111 00:07:35,400 --> 00:07:41,800 Actually a little less with that fudge factor (2 instead of 2.25). But, we get this nice sustained growth. 112 00:07:41,800 --> 00:07:44,438 We can ask, is this going to continue? 113 00:07:44,438 --> 00:07:49,500 Are we going to go from 10 to 20, 30, 40, 50, 60, or is it going to fall off. 114 00:07:49,500 --> 00:07:54,600 But, we see a hint that it is falling off a little bit. This number should not be quite 40. 115 00:07:54,600 --> 00:07:58,400 It should be a little bit less than 40. Instead of 20 to 30 to less than 40. 116 00:07:58,400 --> 00:08:01,100 We should be asking 'are there limits to this growth'? 117 00:08:01,100 --> 00:08:04,400 To try and understand is growth is going to stop or not, let's do the problem. 118 00:08:04,400 --> 00:08:06,800 Let's assume we have a big number of machines. 119 00:08:06,800 --> 00:08:11,500 The number of machines continues to grow unless something different happens. Let's look at 400 machines. 120 00:08:11,500 --> 00:08:17,300 If we have 400 machines, our Output is going to be ten times the square root of four hundred. 121 00:08:17,300 --> 00:08:23,346 So that is going to be ten times twenty which is two hundred ((10)(√400)) or (10)(20)=200 . 122 00:08:23,423 --> 00:08:25,400 So that's great, that's huge GDP, huge output! 123 00:08:25,400 --> 00:08:28,000 What's our Investment going to be? 124 00:08:28,000 --> 00:08:35,200 Well, we had 200 as Output (0.3)(200). Our Savings Rate is 0.3 . 125 00:08:35,200 --> 00:08:39,700 So that means we are going to Invest in 60 machines. 126 00:08:39,700 --> 00:08:52,500 But, Depreciation is going to be (400)(0.25) which is 100 machines. 127 00:08:52,500 --> 00:08:58,000 Investment = 60, Depreciation = 100 (60-100=-40) 128 00:08:58,000 --> 00:09:01,100 So if we started off with 400 machines, we would fall off to 360 machines. 129 00:09:01,100 --> 00:09:03,100 [laughter] So wait a minute! 130 00:09:03,100 --> 00:09:07,400 Somehow this economy is going to grow, but it can't grow this big. 131 00:09:07,400 --> 00:09:11,100 If it grew to 400 machines it would shrink back down to 360 machines. 132 00:09:11,100 --> 00:09:20,700 Is there some number we could reach. We started out with 4. We'd go from 4 to 9 to 16 and so on. It looks like it is never going to stop. 133 00:09:20,700 --> 00:09:23,515 Then we say we have 400 (machines) would continue to grow. 134 00:09:23,638 --> 00:09:26,369 We find out, No, it's not going to grow, it's going to stop. 135 00:09:26,369 --> 00:09:28,158 In fact it's going to shrink down. 136 00:09:28,158 --> 00:09:29,500 There must be some place where it's going to stop. 137 00:09:29,500 --> 00:09:32,200 There must be some natural limit to the growth. 138 00:09:32,200 --> 00:09:35,815 In fact that is what economists like to call "The Equalibrium" 139 00:09:35,815 --> 00:09:40,500 Right. If we look at this growth it is going up, up, up, and it is going to flatten out. 140 00:09:40,500 --> 00:09:43,700 This flat line here is going to be the Equilibrium level. 141 00:09:43,700 --> 00:09:47,500 What we want to understand - what is that Equilibrium level? 142 00:09:47,500 --> 00:09:51,200 Let's think about it. What's going to happen in Equalibrium? 143 00:09:51,200 --> 00:09:54,200 With Equilibrium the number of machines stays fixed. We stop growing. 144 00:09:54,200 --> 00:09:57,000 But, what effects the number of machines? Two things. 145 00:09:57,000 --> 00:10:00,700 One, Investment. You buy new machines based on your Savings Rate. 146 00:10:00,700 --> 00:10:04,600 What else? Depreciation. You loose machines due to Depreciation. 147 00:10:04,600 --> 00:10:10,700 So, the Equilibrium occurs when Investment = Depreciation. 148 00:10:10,700 --> 00:10:13,931 Okay. Well Guess what, that why this is so easy to solve. 149 00:10:13,931 --> 00:10:15,600 That is why models are so great! 150 00:10:15,600 --> 00:10:16,700 Let's do this formally. 151 00:10:16,700 --> 00:10:19,731 So, think about it. What's our Output (Output equals ten times square root of M (Output=(10√M). 152 00:10:19,731 --> 00:10:20,892 What is our Investment? Well, that's easy. 153 00:10:20,892 --> 00:10:28,092 Investment is 0.3 ten times square root of M which equals three times square root of M. ((0.3)(√M)) = ((3)(√M)). 154 00:10:28,092 --> 00:10:30,000 What's our Depreciation? 155 00:10:30,000 --> 00:10:43,200 That's just (M/4). In Equilibrium, (Depreciation) must equal (Investment). So again, easy ((3)(√M) = (M/4) 156 00:10:43,200 --> 00:10:55,700 So that means that ((12√M)=(M)), 12=√M, M=144 157 00:10:55,700 --> 00:11:03,731 So, if my total number of machines are 144, the depreciation is going to be exactly the same as the Investment. 158 00:11:03,731 --> 00:11:15,669 Again, here's the math, Investment = Depreciation (0.25M = 3√M. M=12√M, M=144 159 00:11:15,669 --> 00:11:17,400 Let's go ahead and check. 160 00:11:17,400 --> 00:11:19,000 d=0.25, s=0.3, machines=144 161 00:11:19,000 --> 00:11:45,177 Output=10√144=120 Savings= (0.3)(120)=36 machines. Depreciation is 1/4(144)=36. So, Investment=36 and Depreciation is 36. 162 00:11:45,177 --> 00:11:49,031 The total number of Machines remains at 144. We have reached and Equilibrium. 163 00:11:49,092 --> 00:11:52,900 The number of new machines and the number of old machines cancel out. 164 00:11:52,900 --> 00:11:55,296 That's exactly where that curve finishes. 165 00:11:55,296 --> 00:11:57,331 Output is going to be at 120. 166 00:11:57,331 --> 00:12:06,569 Right? So what we get is a Long Run Equilibrium rather than a Small Run of exactly 120 units of output total. 167 00:12:06,754 --> 00:12:11,685 There's something ironic here. I call this a Growth Model. 168 00:12:11,900 --> 00:12:13,462 Well, what's ironic about this? 169 00:12:13,462 --> 00:12:16,446 What's ironic about this, is that eventually there is no growth. 170 00:12:16,446 --> 00:12:23,344 We're starting with 4 machines, then 9, then 16 then 23, and so on, eventually there are 144 machines, and we stop. 171 00:12:23,344 --> 00:12:24,700 Growth stops. 172 00:12:24,715 --> 00:12:28,215 What's going on? Well, let's think about it. 173 00:12:28,215 --> 00:12:30,146 Depreciation is linear. 174 00:12:30,146 --> 00:12:33,131 So Depreciation is just a nice linear function. 175 00:12:33,131 --> 00:12:37,200 But, our Output is a function of our machines is Concave, thats falling off. 176 00:12:37,200 --> 00:12:42,585 At some point, the amount of more Output that we're getting is falling 177 00:12:42,585 --> 00:12:46,662 to match the slope of depreciation, and those things exactly balance out. 178 00:12:46,662 --> 00:12:48,192 And that's why growth stops. 179 00:12:48,192 --> 00:12:51,700 Well, if growth stops, how do we get more growth? 180 00:12:51,700 --> 00:12:54,185 Well, the answer is Innovation. 181 00:12:54,185 --> 00:12:56,344 Innovation allows us to continue to grow. 182 00:12:56,344 --> 00:12:58,492 That's why people focus so much on Innovation. 183 00:12:58,492 --> 00:13:00,546 So, to get a real model of Economic Growth, 184 00:13:00,546 --> 00:13:03,208 we've got to move beyond this Simple Growth Model and 185 00:13:03,208 --> 00:13:10,240 actually include a 'Parameter' that takes into account technological growth. 186 00:13:10,240 --> 00:13:13,438 So let's step back. What have we learned? 187 00:13:13,438 --> 00:13:16,777 We've learned, that if we write down a Simple Model of Growth - 188 00:13:16,777 --> 00:13:20,646 Economic Growth that involves investing money in new machines. 189 00:13:20,646 --> 00:13:26,500 That there are limits to growth. That the model is going to max out at this point, 190 00:13:26,500 --> 00:13:33,200 when the number of machines lost to depreciation is exactly offset by the number of machines that we invested in the previous period. 191 00:13:33,446 --> 00:13:39,600 If we start with no machines, growth is going to happen really really fast initially, but then it's going to fall off when it reaches this equilibrium level. 192 00:13:40,815 --> 00:13:46,500 So to get sustained growth, that's going to require new technologies - new innovations. And that's where we are going next. 193 00:13:46,500 --> 00:13:52,600 We're going to construct Solow's Growth Model which includes this Innovation Parameter. 194 00:13:52,600 --> 00:13:54,400 [excitement] Yet, even more complected, I know! 195 00:13:54,400 --> 00:13:58,600 By including this Innovation Parameter, we'll see how growth can be continued to be sustained. 196 00:13:58,600 --> 99:59:59,000 Thanks.