The aggregate production function and growth | APⓇ Macroeconomics | Khan Academy

In a previous video, we have introduced the idea of an aggregate production function, which is a fancy way for a mathematical model that an economist might use to tie the factors of production in an economy to the actual aggregate output of an economy. The aggregate output is Y, and then the factors of production— we've talked about this before— it's human capital, it's technology, and it is regular capital or non-human capital.

A is really representing the technology factor here, and this term is often known as total factor productivity. K is referring to the non-human capital, and of course, capital starts with a C, but they use K for capital. Then L stands for the human capital; you could view it as standing for labor, but standing for a little bit more than that, and we'll talk about that in a second.

Just to make this tangible, because it's written in function notation here, which might seem a little bit abstract, it's just saying, "Hey, some function of K and L." You could imagine an aggregate production function that looks like this where our aggregate output is equal to our total factor productivity, which is once again a measure of our technology, times our capital to some power, times our human capital to some other power.

In an introductory economics course, you wouldn't actually have to do this type of computation, taking things to fractional exponents—although we have many videos on Khan Academy explaining how to take fractional exponents if you are curious. But this gives you a sense that, look, if any one of these inputs goes up, well, then you would expect aggregate output to go up, and if for whatever reason these were to go down, then you would expect any one of these to go down—that would have a negative impact on aggregate output.

But the focus of this video is really thinking about, if you were an economist, how would you actually come up with the values for K and L? Pause this video and think about that.

So let's start with L, which sometimes we imagine represents labor, but you really should think of it as human capital. How would you measure human capital? Well, the most obvious thing is you could measure labor. And how would you measure labor? Well, you could go and see, well, how many people are in the labor force? So measure the labor force.

You might say, "Well, isn't that all there is to human capital?" But remember, not all labor is equivalent. If people are unhealthy, they're not going to be able to output as much. If people aren't trained, they're not going to be able to output as much. So it's not just the quantity of people in the labor force; it's also measures of education that would factor into L. So the more educated a labor force, the more trained L would go up. Higher people in the labor force means L goes up; a more educated labor force means L goes up, and a healthier labor force means L would go up—they're going to be able to do more.

Now, what about K? What about the capital stock of a country? Well, you might be tempted to say, "Well, maybe I could count the amount of capital or something like that," but that wouldn't really make sense because there could be some types of capital that might be, you know, some just tools, while you might have another capital that's a big building, a rail car, or whatever else.

So the K is actually measured by economists as the value of the capital stock in a country. So let me write it this way: the value of capital in your country or in the economy that we care about.

Now, a really interesting one is how do you measure A? There isn't an obvious index for, "Hey, I can just observe that and say that has more technology than that other thing." The way that economists often figure out an A for an economy is by backing into it. They can figure out an L, and they can figure out a K, and they know what the output is; they know what the GDP of that country is, the real GDP.

If different countries have the same K and the same L, but then their GDPs are different, then that means that they have a different A. So the A you could almost view as an adjustment to fill in the gap to connect the dots between K, L, and Y. But it would be a measure of how technologically advanced something is. If my economy has an A of 1, in order to make the numbers work, and your economy has an A of 2, that means for some reason you're getting twice the productivity given the same capital and human capital as I am, which implies that you have twice the technology.