Cobb-Douglas pt. 1? AKA: Introduction to brain pain.

Pretty colors.

Alright, since I'm contemplating making a Youtube account to create tutoring videos, I might as well test the waters here. Since I haven't mentioned math in about two posts, guess what time it is? Don't worry, I'll keep it simple.

I want to cover the simple Cobb-Douglas production function. I'll lay it out, and then go about explaining it. Generally, it's written as:

Which is relatively tame once you know the purpose and definitions of the terms. L is labor, K is capital, A is known as "Total factor productivity", Y is total production, and the exponents I'll explain in detail later.

As for now, notice the structure. Total production, Y, is dependent on how much labor and capital, L and K, you put into it. Formally, it's the dependent variable in Y=f(L,K). Now, notice that when the exponents are greater than zero (A must also be positive!), then adding more labor or capital into production increases the amount produced. Obviously, that makes sense. As for the coefficient, A, that's basically a measurement of how productive and technologically advanced the production process is. Here's what Wiki says about it:

"If all inputs are accounted for, then total factor productivity (TFP) can be taken as a measure of an economy’s long-term technological change or technological dynamism."

If TFP doesn't make sense, don't worry, I'll go over it later in probably a different post. 

Now, here's something that, at first, will seem complex but is incredibly simple once you get used to it. In economics there are the concepts of "increasing returns to scale", "constant returns to scale", and "decreasing returns to scale". Think of them like this. Let's say you own a business, and you want to increase production. Let's say you double all inputs of production. What happens? If you more than double output, then you're in the "increasing" section. Adding one unit of input, gives you more than one unit of output. Let's say doubling the inputs gives you double the outputs. Well, then you have "constant" returns to scale. Adding one input gave you exactly one output. What about the final scenario, namely, you doubled the inputs but got less than double the output? Well, using the process of elimination (or your brain stem) would tell you that would be the "decreasing" returns to scale. Well, let's try it with our function. Let's say we increased all inputs by t. That would be the same as Y=f(tL,tK). Let's try that:

In other words, increasing each input by t, increased the output by t^(α+β). You can see this because increasing all the inputs by t is equivalent to multiplying the total production by t^(α+β). What does this tell us? Well, if α+β>1, then you have INCREASING returns to scale. To see this, it means that increasing inputs by t, gives MORE than t output. If α+β=1, well, then we have constant returns, because t^(α+β)=t^1=t. In other words, increasing by t caused production to increase by t. What about the final scenario, α+β<1? Well, as you'd guess, that's decreasing returns to scale, because increasing by t causes production to increase less than the increase in inputs.

That's about half the battle. Only one other thing that I want to cover, and that's marginal productivity. Marginal productivity is about what you'd expect. How much does increasing an individual input (while holding the other constant) by a small, marginal, amount increase total production? Notice how they're different. This involves increasing individual inputs. Here we can make good use of the partial derivative, and it gives:

Where MPK is the "marginal productivity of capital", and MPL is the "marginal productivity of labor". In other words, giving a small increase of either, those equations give the small increase in output.

Probably too mathematical for this crowd. Still, a good place to start, but I'll probably pick up more of this stuff later. Back to non-rigorous plebeian stuff for a little bit.