Hi. So today we begin our first lecture on the Solow Model. This is a workhorse model in economics, not just for understanding growth, but also in further developments for understanding business cycles. The model was developed by Robert Solow and he won the Nobel Prize in 1987, largely for his work on this model. In this lecture, we're going to begin with the super simple Solow Model, which is the version that Tyler and I developed in our textbook "Modern Principles". So if you want to follow along, I'd get a hold of that textbook. The Solow Model begins with the production function, which is simply a mathematical model describing how output is produced. So we're going to write output 'Y' as a function of physical capital 'K', human capital, which we're going to write as 'e times L'. You can think of that as education times the number of laborers. Ideas, which we're going to write as 'A'. Later on, we will also interpret this 'A' factor as productivity. So we're going to write output is equal to a function of ideas, physical capital and human capital. Now to begin, we're going to assume that 'A', 'e', and 'L' are constant. So there's no population growth, there's no idea growth, there's no education growth. This is nice because it means that we can then write: output is simply a function of capital. Now, what kind of function is this? What kind of property do we want this function to have? Basically, there are two. Clearly, we want more capital to produce more output, a positive function. But we want it to do so at a diminishing rate. That is, each addition to capital should generate smaller additions to output, the more capital we have already. I'll explain that a bit more in a second. A nice simple function which has these properties is the square root function. So we're going to write: output is equal to the square root of capital. For example, if there are 16 units of capital, then output is 4. Let's describe this function in more detail. Okay, so here is our production function. We have capital along the horizontal axis and output along the vertical axis. And notice that at a capital stock of 100, you get an output of 10, the square root of that. At a capital stock of 400, you get an output of 20. Okay, notice also that this graph has diminishing returns. Let me show you what I mean. Let's suppose this is a farm, a wheat farm, and let's imagine that a tractor is 100 units of capital, this is arbitrary. Your first tractor, that helps you to produce 10 units of output. Your first tractor helps you to produce 10 units of output. The second tractor's not quite as useful as the first tractor because you've already got one tractor running. So the second tractor you can't run quite as often. You only get about 4 or 4.1 units of output from your second tractor. What about the third tractor? Well, the third tractor you can only use, let's say, when the first two tractors break down. So on average, your third tractor produces even less than your second tractor did. In fact, adding a third tractor now gets you only about 3 or 3.1 units of output, 3.2 or something like that. Okay, so this is our production function. It has diminishing returns to capital. We've only just begun to develop the model but already there are some hints and some ideas we'll be developing in future lectures. In particular, due to diminishing returns to capital, countries with small capital stocks should grow rapidly. So you take a country where the capital stock is really low and that means that the productivity of capital ought to be high. So even a little bit of investment ought to generate you a big increase in output, a high growth rate. So China may be one example of this. So, in China they are almost literally, literally in fact, adopting the first tractors. Those first tractors are increasing agricultural output tremendously, which is a high growth rate. In fact, across all kinds of sectors of the economy, China is adding its first units of capital and that is growing the economy rapidly. What this does also suggest, however, is that China is going to slow down as capital accumulates. Now, if it's the case that countries with small capital stocks should grow rapidly, why don't all poor countries grow rapidly? If you look around the world, it's just simply not the case that poor countries on average grow faster than rich countries. In fact, the reason they're poor is often that their growth rate is low. So why aren't all poor countries growing rapidly? Well, what China did is adopt more capitalist institutions. It freed up its property markets. It improved incentives, you know, after the death of Mao, in particular. And it was only after it improved its institutions, that it was able to take advantage of the high productivity of capital and to grow rapidly. So, here we've got a hint at something else we're going to be talking more about, conditional convergence. The countries can grow rapidly towards what we might call their natural GDP per capita, what we might call the GDP per capita, which is sort of fundamentalist, comes out of the fundamental quality of their institutions, the fundamental savings rate, and so forth. Things we'll be talking about more later. Okay, here's another interesting application. Bombing a country, can increase its growth rate. Let's look at some data. So this is data from Germany, Japan and the United States. Let's start with 1950 to 1960, the decade after World War II. What do we see? Well, we see Germany growing at a 6.6 percent growth rate, Japan at a 6.8 percent growth rate. Meanwhile, the United States is only growing at 1.2 percent. What is going on? How is it that the losers of World War II are growing more rapidly than the winner? To a lot of people this just didn't seem right. But actually, the explanation is pretty simple. The explanation is that huge amounts of the capital stock in Germany and Japan were destroyed. So Germany and Japan had a low capital stock but their fundamental convergence GDP per capita rate, the rate towards which their fundamentals were leading them was high. So their capital stock indeed was very productive. They were able to invest a lot and growth rates were very high. However, as capital accumulated their growth rates fell. So Germany fell to 1.9 percent from 1989 to 1990, Japan to 3.4 percent, and of course, in the next decade it slowed down even more, the United States at 2.3 percent. This also suggests another idea we're going to be talking more about in future lectures. You see if these high rates, 6.6 percent in Germany and Japan and so forth, China's 10 percent rate, if these high rates are all about catching up, all about really high productivity capital, where you're catching up to your natural level of GDP per capita. You know, what is determining this rate, where you're at your sort of natural GDP per capita rate? What is determining growth on the cutting edge? We'll be talking more about that when we talk about ideas in a future lecture. Okay, let's get back to developing the model. So, we've talked about how capital produces output. Now we want to say, well, what do you do with that output? We're going to split it into two basic things. You can invest the output, or you can consume the output. So, we have a very simple and good model here. Think about potatoes, you know you produce potatoes, that's your output. You can invest. You can take some of those potatoes and plant them back into the ground. That's an investment of potatoes in order to produce more potatoes in the next period, or you can consume the potatoes. So those are the two things we're going to let output do. We're going to assume that people take a constant fraction of output and save and invest that constant fraction. So here is our investment curve, the green line. Investment, in our case, is going to be equal to 30 percent of output, so a savings rate of 30 percent. In other words, our investment curve is equal to 0.3 times "Y". So we take here as our output curve and we multiply that by 0.3. That gives us our investment curve. In other words, investment is equal to 0.3 times the square root of "K". Let's take a given capital stock. So if we have a capital stock of 100, that means that output is 10, the square root of that. And of those 10 units of output, three units are going to be invested and seven units are going to be consumed. Pretty simple. The next thing we need to add to our model, the next fact we need to take into account, is that capital depreciates. So those tractors begin to rust, they begin to fall apart. Think about a car. You bring home a new car. It runs perfectly but after a while it needs more maintenance. it needs more oil checks. The brakes start to fail. Things start to fail. You have to invest money just to keep the car running the way it was when you bought it as new. Factories begin to depreciate. There is wear and tear on the factories. Water systems, sewer systems begin to corrode. They begin to fall apart over time. And you need to invest just to keep you at the same place that you were before. So this is depreciation, this is capital depreciation, wear and tear. Here's how we add depreciation to our model. Depreciation, we're going to write as just a constant fraction of the capital stock. So we'll write that every 100 units of capital you have, two units depreciate every period, so a depreciation rate of 2%. This means that every 100 units of capital, you have two of them wear out every period. Now, what's going to turn out to be very important, is the relationship between the investment function, in green, and the depreciation function here in siena. In fact, we want to focus in on this because this is the whole model. A lot of the model will be driven by this. Let's focus in on this area. We're going to blow up these two curves so we can see what's going on a little bit more clearly. Okay, here's our investment curve and our depreciation curve, exactly as we had before. I've just blown them up so they're easier to see. Now, here is the key to the model. When investment is bigger than depreciation and when you are investing more capital than is depreciating every period, then your capital stock must be growing. Similarly, when depreciation is bigger than investment, when more capital is depreciating every period than you are investing, then your capital stock is getting smaller. So the capital stock is growing anywhere in this green area over here, anywhere in this green area. The capital stock is getting smaller anywhere in this red area. So the capital stock is shrinking. If capital stock is growing over here and shrinking over here then the one place where the capital stock is constant is where the investment is equal to depreciation. This point is the steady state, where investment is equal to depreciation. The capital stock is holding constant. It's neither growing nor shrinking. The capital stock is constant when you are investing, every period, just enough to replace the capital which has depreciated. That's the steady state, when you are investing just enough to replace the capital which has depreciated. Okay, now we've brought the output curve back in, so we can see the whole model in one diagram. Let's remember that the steady state is when investment is equal to depreciation. That turns out to be a capital stock, where the capital stock is 225, right here. The investment equal to the depreciation is 4.5. That's given right here. Then the output, the steady-state output, is equal to 15. That is when the capital stock is 225. The square root of that is 15. We can think about this, by the way, as GDP per capita. So, the Solow Model is telling us a couple of things. First of all, it's a model of GDP per capita. It tells us, we haven't gone into this in great detail, but you can see it will have something to do with the savings rate. The savings rate is going to be higher. You can see this curve is going to shift up. We're going to get a bigger steady-state output. We'll talk more about that in a future lecture. You can also see that this tells us something about the growth rate, that you're going to grow when you're below your steady-state capital stock. Finally, it tells us that that growth is going to taper off, that growth is going to slow down. You're eventually going to come to a point where the investment is equal to the depreciation. That is, your capital stock is going to grow so large, such that at some point all of your investment is is going to be taken up just maintaining the capital stock. So, you're going to build so many aqueducts, think of Rome, so many sewers, so many water supplies, so many factories that every period the depreciation on those bits of capital is going to be so large that it's going to take up all of your investment. And that's when growth stops. So this model is telling us something, which at first is a little bit surprising. It's telling us that capital growth alone, capital deepening, cannot be responsible for long-run growth. At some point your capital stock will be so large that it's all going to be, all of your investment is going to be consumed by making up for depreciation. So this model is telling us we're going to have to look somewhere else to understand long-run growth. And of course, when we get there, we're going to be talking about ideas. We're going to put the model through its paces in the next few lectures. Thanks.