Transcription
Welcome to this rapid review on the Cobb-Douglas production function. We'll start by writing out the general formula for the Cobb-Douglas production function. It's A times K (capital) to the exponent alpha times L (labor) to the exponent (1 - alpha). K and L are variables; they're capital and labor. As for A and alpha, we're less familiar with A. A is a parameter that basically just represents our controls productivity; basically, the higher A is, then the bigger the total output will be, and we can see that from the formula. Alpha is another parameter, and we call it the capital's share, which is kind of a strange name, but by the end of the video, it'll make sense. We'll come back to this, explaining why alpha is called the capital share parameter.
The first question we can ask about Cobb-Douglas is: Is the Cobb-Douglas PF constant returns to scale (CRS)? Constant returns to scale, and remember the general test is a constant returns to scale production function should give us: if we multiply Z K and L by Z—so we, for instance, doubled them, we multiplied them by 2—then our output should increase by a factor of Z. So let's see if that's true. We just plug in ZK for K; we'll get ZK to the alpha. I'll plug in ZL to the (1 - alpha), and we'll try to rearrange terms, see if this collapses back down to Z times the original production function. So we'll split this out; we get Z to the alpha K to the alpha. That's just our standard rules for exponents. Z to the (1 - alpha) L to the (1 - alpha). Now let's group all the Z's together at the beginning, so we'll have Z to the alpha times Z to the (1 - alpha), and everything else at the end: A times K to the alpha (1 - alpha). And we can see the last three parts are our original production function. This is just our original F(K, L). So hopefully, this part at the beginning here is Z to the alpha times Z to the (1 - alpha) will reduce down to Z, and we can see that that's true, right? When you multiply two things, then you add the exponents, so we'll have alpha + 1 - alpha, so those cancel, and we're just left with Z. So Cobb-Douglas is indeed constant returns to scale: F(ZK, ZL) = Z * F(K, L). So it works out perfectly.
Now we're going to investigate the marginal product of labor and capital for the Cobb-Douglas. This is the part that requires a little bit of calculus, but I think it's instructive. It shows you how to get these marginal products, and we're going to need these marginal products for our third calculation that we do. So first, we'll look at the marginal product of labor. In general, that's dF/dL for the production function, and we'll calculate that for the Cobb-Douglas. The A and K parts don't have an L, so we can just leave those alone, and then we want to calculate the derivative of the L part. So that works out to be (1 - alpha). The exponent comes down, and then we subtract 1 from the exponent, so (1 - alpha) - 1. We can do some simplifying here because there's a -1 here and a 1 here, so this reduces to—well, actually, let's erase this—we'll put the alpha, the (1 - alpha) part on front, and then we'll have A to the K alpha L to the -alpha. And this is fine if we just left it like this. I think this is the way you'll see it in most textbooks, but we can group terms a little bit. I think it's instructive, so we'll have (1 - alpha) times A, and since the K to the alpha and the L to the -alpha essentially have the same exponent, we can group them together by the prior rules for exponents, and we see it's (K/L) to the alpha. So it's like a ratio of our capital to labor. That's kind of interesting. We'll come back to that point in a second, but first, let's calculate our marginal product of capital. This is, in general, dF/dK. Just like before, we'll have the A and the L parts are not affected in this derivative. That will have the exponent on capital come down, so that's alpha, and then we'll have K to the (alpha - 1) because you subtract 1 from the exponent when you do the derivative. We'll group terms again, putting the alpha and A up front, and then we'll have K to the (alpha - 1) L to the (1 - alpha). And just like above, if you use the sort of rules for exponents, so you can see this is—we can write this in terms of a ratio of labor to capital—it's (L/K) to the (1 - alpha). And what I want you to notice now, looking at the two expressions, is that the MPL will increase as you give more capital. So K up gives us higher marginal product of labor, right? Because when we look at this ratio here, as the ratio of capital to labor goes up, our marginal product of labor goes up. So higher capital, higher marginal product of labor. Similarly, as L goes up, we get a higher marginal product of capital because we see in the marginal product of capital equation it depends on the ratio of L to K. So this is nice to notice because it tells us that in the production process, K and L are complements; they sort of work together. The more capital you have, the more productive your laborers are at the margin; the more laborers you have, the more or better you can get out of the capital, the higher the marginal product of capital is. Sometimes people think of production functions as being like: you would use capital or labor. Do you think about pretty much any business process you could imagine, like say running a restaurant? You couldn't do it with just people and no ovens and no pots and pans, nor could you run a restaurant with just ovens and pots and pans and no cooks or servers or anything. You need both the capital and the labor to work together in the production process, and the Cobb-Douglas nicely reflects that.
The last calculation we're going to do is look at the labor share of income and capital of income. So labor share, in general, is the amount of income that goes to labor in real terms over total income, so Y. So labor share would be the real wage paid each labor times the number of laborers over the real income, and then we know the real wage is equal to the marginal product of labor. So this we could substitute and say this is the marginal product of labor times L over Y. Now we calculated the marginal product of labor previously in the video, so we're just gonna borrow the equation for that from our earlier work. It was (1 - alpha) times A times K to the alpha times L to the -alpha, and then we'll have times L, and then Y, we're gonna expand out and write in terms of our like general production function, right? We know Y, we put a note over here: Y = F(K, L), and then we know our production functions to Cobb-Douglas, so Y is A times K to the alpha times L to the (1 - alpha). And now we can see some terms are starting to cancel. The (1 - alpha) is leftover; it doesn't cancel, but the A's cancel, the K's cancel, and then we have some stuff in terms of L, so on L, so the (1 - alpha) on the bottom, and then we have L to the -alpha, and when you multiply, you add exponents, so this is an L to the (1 + -alpha). This is -alpha + 1, and what you sort of notice from looking at this is that this is L to the (-alpha + 1) over L to the (-alpha + 1), basically, right? These are the same thing, so we can cancel these guys, and we're left with (1 - alpha). Labor share of income is always (1 - alpha), and it's constant; it doesn't depend on the level of K or of L; it's just kind of fixed; it even doesn't depend on A. Now we know labor share and capital share add up to 100 percent, so if we wanted to, we could sort of skip that calculation for capital share and just say, well, they must add up to 1, so capital must get alpha. But let's do the calculation just to make sure we didn't make any mistakes. So capital share, which in general is the real rental price of capital times the amount of capital over total income, and that is marginal. We know the real rental price of capital is the marginal product of capital, so we can substitute it. Now we already calculated the real—I'm sorry—the marginal product of capital, so we'll plug in for that. It was alpha times A times L to the (1 - alpha) times K to the (alpha - 1), and then times K here, all over Y. We know Y is equal to our production function, so we'll just plug in the Cobb-Douglas production function. And now we can see, like above, there's a lot of canceling to do. The A's cancel, the L terms cancel. This is a K to the 1, so this becomes alpha times K to the (alpha - 1 + 1) over K to the alpha, and you can see that K to the (alpha - 1 + 1), that that part cancels, right? -1 + 1 just gives you 0, and then we have K to the alpha over K to the alpha, so those cancel, and we're left with just alpha. So capital share is alpha, and this helps explain why we had that name; hence the name.
So let's summarize. With our Cobb-Douglas production function, we noted three important things: first, it is constant returns to scale; second, K and L are complements; and lastly, capital and labor each have constant factor shares. So we could just summarize that as: factor shares are constant. Thanks for watching.