Hi, we just took in a very simple growth model and in that growth model we saw that well, growth stopped, right. Once we got to 144 machines and an output of 120, we no longer got any growth. So we use that very simple model to get at. A really important fact, that without innovation, if technology stays fixed, growth will stop. Now, sure the labor supply could get bigger, we could have more workers or something like that. But holding the amount of labor fixed and holding that technology fixed, if we've got a fixed savings rate, and a fixed rate of depreciation, there's no more growth at some point. We're gonna go up, up, up, up, up, and then stop. Well. That hasn't been human experience right. Economic well being continue to go way up right and GDP continues to go up, and so what's driving that. Well to get at that we're gonna look at a deeper model, richer model known as the solo growth model. And what's nice about this model and I love about this model, we're just gonna add one variable. We're just going to add one more variable to our other model, and that's suddenly going to give us a way to include innovation. Now, just to make this, you know. More interesting [laugh], maybe more real. These models are developed by real people. Right, and so, the speaker model's developed again by Bob Solo. And Bob Solo is an economist at MIT. And here's Bob right here. And this is Bob actually testifying before the House Science Technology committee on the need to have multiple models to understand the economy. Right, so this is a group of economists here and we're standing up and we're swearing to tell the truth, the whole truth, and nothing but the truth about why models are important to understand where growth comes from. And in this particular case, to prevent things like, you know, the home mortgage crisis, which cost us all a lot of money. Alright, so how does Solo's model work? What does Bob's model do? Well, what Bob does is this wonderful thing. He includes. Includes one more variable. So everything's the same as before, [inaudible] labor, capital, depreciation, savings. But we're gonna include this thing A of T which stands for technology. So when A is low technology is low when A is high technology is high it's better so A's just going to be the parameter we can tune to effect sort of how much or how good is the technology in the economy so for making cocoanut picking machines [laugh] right A is really small and if we're making incredibly cool you know laser pointers and IPhones and stuff like that, technology is great. Okay, so this is it. Very simple formula. Output is just equal to the technology at that time, times capital to some. Beta. And L to sum one minus. Now wait a minute. This also got a little more complicated. Now I got these betas here. Now before I had square roots. Well, if beta equals one half. Right, so beta is a half. Then this is just the square root of labor times the square root of capital. Right? Easy. If beta gets bigger than a half, that means that capital matters a little bit more. If beta gets less than a half, then that means that capital matters a little bit less. So depending on the technology, it could be that it's a capital intensive technology so that beta would be big. Doesn't use that much capital and beta tends to be relatively small. So, you can estimate different [inaudible] betas for different manufacturing processes or even for different countries, right? And a half was just a convenience we assumed. So that's actually something that we take models to beta, you go and estimate and figure out what is beta. And for us, if we're just trying to get the ideas here, right? And we're going to take beta equals a half. So let's go back, and just to remind ourselves of where we were before, right? Remember our total output was ten because we [inaudible] a hundred workers, so ten times the square root of N. We had a savings rate of 30%. And we added a depreciation at a quarter. And we went through and we did all that stuff we saw the equilibrium where the investment was exactly equal the depreciation. When we got to that happened we put an output of 120 which required 144 machines, right? So that meant that we were going to invest in 36 new machines but we'd lose 36 machines to depreciation. So that was our equilibrium. Now we want to say well, what would innovation do? Well, innovation would do, was, would we'd put an A in front of this. Now have an A in front of this ten times the square root of m. So, let's do that and let's see what happens. So now we're going to say the output it. Two times ten times the square root of eleven. So what we're going to do is, we're going to assume that somebody had a technological innovation and our coconut machines are now, somehow like, everything's twice as good, or twice as productive. Okay, well now let's, let's walk through the math. So, what's our investment gonna be? Investment is gonna be 0.3. Times twenty the squared of M. So that's gonna be six times the squared of M. And what's our depreciation? Well, that's gonna be one-fourth M, right? So that's just M over four. And so we just have to set these things equal again, right? So six squared of M equals M over four. So that means 24. Square root of M, equals M. So that means 24, equals square root of M. So that means M equals. Right? So our equilibrium is gonna be m is equal 496. And output, right, is gonna be two times ten times the square root of 496, which is 24, right? So that's 24 times ten which is 240, which is 480, right. So our output 480. Before it was 120, and now it's 480. So, think about it. Productivity doubled, right, [inaudible] our technology got twice as good. But long run GDP went up by four. But why is that, well let's look back at our numbers here. Okay, we became twice as productive so that means if we had kept the number of machines at a 144, we now would have an output of 28, 240. Right, so we would have doubled where we were at. But we didn't. Keep the number machines at 144 when the technology are better we actually increase the number machines to 496. So when you have a technological change two things happen. First you just get more productive you get more stuff, second because you?re getting more stuff it makes sense to invest in more machines. So there's this multiplier effect so that means productivity goes up by two right. Output eventually the long run, long run [inaudible] goes up by four right, and this is what you think of as sort of as a innovation multiplier because it happens were there's these two effects, right. Labor and capital become more productive. So that, boom, you just get more stuff. But second of all, because they're more productive it makes sense to invest in more machines so then you get even more stuff. So there's this multiplier. Well, let's think about it. Productivity are up by two. Total it up. Eventually in the long run, not immediately. Gotta build up all of those machines. It goes up by four. Well that leads to a puzzle and here's where models are really useful. Is it additive, or is it multiplicative? Here's the issue with two. It could be that productivity went up by two. And so we get two plus two, and, so [inaudible] by four. Or it could be we get 2X2, two squared is the reason productivity went up by four. So we wanna figure out, is this additive effect, right? The machine effect plus the productivity effect. Or is it multiplicative, is it 2X2? Well, to make sense of that, what we can do is, we can increase the multiplier to three. Because if it's additive then we get six, and if it's multiplicative, we get nine. Right so if we make this three times as productive we're going to ask in the long run do we end up with six times as much stuff or do we end up with nine times as much stuff and again this is [inaudible] why do we model we model to get the logic right okay without the model. It'd be very hard to figure out, is this gonna go by six, or is this gonna go by nine. Heck, we might not have even have got the second effect of more machines. Right, so the model was only useful just to giving us that. Now it's gonna tell us the magnitude of the effect. 'Kay, just to get our bearings again, let's remember where we started from. We start from an existing technology where it's just the square root of labor times the square root of machines. We see 100 units of labor. So it's ten times the square root of M. We save 30 percent and invest that in new machines. Right, so that's gonna be three times the square root of M. We lose a quarter of our machines to depreciation, so that's just M over four. We set those things equal. We get m equals 144, we get output of 120. That's our equilibrium. Now we wanna say lets triple it okay so let?s suddenly assume there's an A that comes in that's got a value of three, so now we're going to get three times ten times the squared of M [inaudible] 30 times the squared of M and let?s see what happens okay so what's our total investment going to be? Well that's going to be 0.3 times 30 times the squared of M so that's going to be nine. Times the square root of M. What's our depreciation? Well, that's still just M over four. So let's set these equal. Nine times the square root of M equals M over four, so that means 36 squared of M equals M. So that means 36 equals the square root of M. So is equal to 36 squared, right? So M, we just keep it as 36 squared. Right? So N equals 36 squared. What's total output gonna be? So if we've got 36 squared machines, which is a big number, what's output gonna be? Well, output is three times ten times the square root of 36 squared. So that's three times ten times 36. Well, three times 36 is 108, so that's 1,080. So what happened when we made ourselves three times as productive? Well. Total output went up nine times. So what we see, remember what was our question? Our question was, is it additive? Are we gonna get three plus three are six. Or multiplicative, three times three are nine? The answer is, it's gonna be multiplicative. We're gonna three times three is nine. Two effects multiplied on top of one another. Right, the first one is we just get more stuff. The second one is we invest in more machines. And those two effects get multiplied together. So becoming three times more effective means we get nine times in the long run equilibrium. So let me summarize for a second. In the simple growth model, growth stopped. Right? At some point we got to 144 machines and then we no longer had any growth. When we go to the Solow growth model, what happens is, if we can continue to increase that A, right? So, if we can continue to increase our productivity, then growth can continue. That sort of begs the question. Where do increases in A come from? And this has led to what people call endogenous growth models. So an endogenous growth model, labor can go to things like picking coconuts. Labor can also go to things like, investing in new technologies, research and design, and those sorts of things, to try and increase that A parameter. So what we can think of is before all of our labor went to increasing cap, picking coconuts, right? Now, that labor could also go to doing research on new coconut picking machines. What you get in indigenous growth model is how much labor goes into actually making stuff, and how much goes into research and design and thinking, right. It's a choice variable. In the model, right, and you solve for how much of that you get. Quick summary, right. Growth ceases without innovation, if that's true. Everybody should be pro innovation. And in fact, most people are. Right, so here's two quotes. Here's a fun little quiz. One of these quotes is from President Obama, who is a democrat. The other quote is from President Reagan. I want you to try and guess which quote came from Obama, and which quote came from Reagan. All right? So both Reagan are pro innovation, right? They, they are, because they are pro GDP growth, because [inaudible] gonna lift those people out of poverty. Right? We wanna make everyone better off. Because if we can lift people out of poverty, we make people happier. And what we've learned is the way to do that, right, is. First, by investing in capital. Right? Because that makes us, you know, all do better. We get to the 144 machines. But at some point then growth stops and then you need investment in technology. Then investment in technology leads to innovation which raises the whole thing up and we get this multiplier effect. Right? We get the increased productivity and then we also get the incentives to produce more machines. Right? Which raises our statement [inaudible] even more. So that's the E, in essence what growth theory is. Thanks.