How economists came up with the Cobb Douglas function? How can I deduce it by myself?

Question

In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly physical capital and labor) and the amount of output that can be produced by those inputs. The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1927–1947.

How we get the formula $f(K,L)=AK^aL^{1-a}$? How we get that $K$ must be elevated to some factor "$a$" and this must be multiplied by $L$ elevated by some factor "$1-a$"? What is the proof of this formula? I can't find an answer to this questions.

Update. I found an article that shows how to came up with it, the idea of the Cobb Douglas function cames from certain assumptions about the production function that I will show later.

The problem is that I don't understand some steps, could someone help me? Here goes the link:

https://www.studocu.com/en-gb/document/kings-college-london/mathemtics-for-economists/lecture-notes/cobb-douglas-revision/4229598/view

In these terms, the assumptions made by Cobb and Douglas can be stated as follows:

1. If either labor or capital vanishes, then so will production.
2. The marginal productivity of labor is proportional to the amount of production per unit of labor.
3. The marginal productivity of capital is proportional to the amount of production per unit of capital.

Solving. Because the production per unit of labor is $\frac{P}L$ , assumption 2 says that:

$$\frac{∂P}{∂L} = α\frac{P}L $$

for some constant α. If we keep K constant($K = K_0$), then this partial differential equation becomes an ordinary differential equation:

$$\frac{dP}{dL} = α\frac{P}L $$

This separable differential equation can be solved by re-arranging the terms and integrating both sides:

$$\int \frac{1}P \, dP = α\int \frac{1}L \, dL$$ $$ln(P)=α*ln(cL)$$ For example here, from where it comes the constant "c"?, then following: $$ln(P)=ln(cL^α)$$ $$P(L,K_0)=C_1(K_0)L^α$$ where $C_1(K_0)$ is the constant of integration and we write it as a function of $K_0$ since it could depend on the value of $K_0$.

Answer 1

The exponents of K and L add up to 1 because we assume constant returns to scale. Thus if K and L increase X% then output increases X%.

Using some math (which I forget) we can show that the exponents $\alpha$ and 1-$\alpha$ represent the earnings share of K and L. So (from the income approach) if 0.3 of GDP goes to capital and 0.7 of GDP goes to labor, then $\alpha=0.3$

Answer 2

To answer your question about the article and the constants, I believe that is a typo and they meant to make it an adding of a constant. The idea should be that you integrate both sides of $$\frac1P\,dP = \alpha\frac1K\,dK$$ to get $$\ln P = \alpha\ln K + C = \ln K^\alpha + C,$$ where $C$ is your constant of integration. Then when you take the exponential of both sides, you get $$P = e^{\ln K^\alpha + C} = e^{\ln K^\alpha}e^C = K^\alpha e^C.$$ They then set $C_1 := e^C$ (since $C$ is a constant, so is $e^C$) to arrive at $P = C_1K^\alpha$, where $C_1$ might depend on the $K$ that we fixed (i.e. $K_0$, hence why they wrote it as a function of such).