The Cobb-Douglas Production Function

Heads up — this one’s gonna be pretty short.

Promise.

So that vague little function I keep waving around,

$$f\left( \frac{K}{N} \right)$$

let’s actually pin it down.

The one people actually use in practice — apparently — is the Cobb-Douglas production function.

Back in 1928, Charles Cobb (mathematician) and Paul Douglas (economist) cooked up an

$$f\left( \frac{K}{N} \right)$$

that fit the economy of their time pretty nicely~

…and it stuck.

Here’s what Cobb-Douglas looks like:

In other words,

$$\frac{Y}{N} = \left( \frac{K}{N} \right)^{\alpha}$$

That’s it!!!! (Basically just $f(x) = x^{\alpha}$, right?)

Now bring back the steady state from last post —

$$\frac{Y^{*}}{N} = \left( \frac{K^{*}}{N} \right)^{\alpha}$$

That’s the steady-state version.

And what we want to do is take that $Y^*$ — the output when growth is zero — and write it out using the actual production function we just plugged in!!!!

OK so the steady state is set by

$$s \cdot f\left( \frac{K^{*}}{N} \right) = \delta \cdot \left( \frac{K^{*}}{N} \right)$$

Plug in Cobb-Douglas:

$$s \cdot \left( \frac{K^{*}}{N} \right)^{\alpha} = \delta \cdot \frac{K^{*}}{N}$$$$\left( \frac{K^{*}}{N} \right)^{\alpha - 1} = \frac{\delta}{s}$$$$\frac{K^{*}}{N} = \left( \frac{\delta}{s} \right)^{\frac{1}{\alpha - 1}}$$

Now raise both sides to the $\alpha$:

$$\left( \frac{K^{*}}{N} \right)^{\alpha} = \left( \frac{\delta}{s} \right)^{\frac{\alpha}{\alpha - 1}}$$

And the left side — that’s our steady state output per worker under Cobb-Douglas. So:

$$\frac{Y^{*}}{N} = \left( \frac{\delta}{s} \right)^{\frac{\alpha}{\alpha - 1}}$$

Way more concrete than last post, sure. But…

we don’t know $\alpha$?!?!?!? How are we supposed to nail down $\alpha$??!

Turns out — if you just rough-fit it against actual economic data,~

When $K$ is treated as physical capital only, apparently $\alpha = 1/3$.

Plug that in →

$$\frac{Y^{*}}{N} = \sqrt{\frac{s}{\delta}}$$

When you let $K$ cover physical and human capital, apparently $\alpha = 1/2$ is the right call.

And then →

$$\frac{Y^{*}}{N} = \frac{s}{\delta}$$

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Originally written in Korean on my Naver blog (2016-01). Translated to English for gdpark.blog.