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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 i search it a lot.

Update. I found an article that shows how to deduce it, but 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 $K_0$.

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What is the proof of this formula?

There is actually no proof for what the production function should be. There are infinite many possible production functions and to discover which one is the most appropriate we need to make some empirical observations. In different cases different production functions are appropriate. Cobb-Douglas is popular production function but I seen many others as well.

What you provide below in your update is not so much of a proof that production has to be Cobb-Douglas rather its a proof that if we make some specific assumptions about production (which although quite general might not always necessary hold actually) we get function that is Cobb-Douglass.

$ln(P)=α∗ln(cL)$ For example here, from where it comes the constant "c"?,

the $c$ is an integration constant. Whenever, you have indefinite integrals you have to add some constant $c$ to the solution because constants get eliminated during differentiation we can never know if there was or wasn't some constant previously so after integrating we always add $c$.

In this case when you integrate this separable differential equation the solution would actually look like:

$$\ln(P)= \alpha (\ln(L) + C) \implies\ln(P)= \alpha \ln(cL) | C= \ln(c) $$

(actually the variables should even be in absolute values - but since the function is only defined for non-negative values of $P,L$ and $K$ we can omit them).

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  • $\begingroup$ Nice, i'm starting to understand, i have another question: why he asume that constant "c" must be a function of $K_0$? $\endgroup$ – Sandro Oct 2 '20 at 20:02
  • $\begingroup$ @MarceloEncisoJure actually it’s not appropriate to say a priori it must depend on $K_0$ but in this case given the conditions laid out in that document, especially that the production depends on both factors and when one factor is zero production is zero we can reason it constant will be function of $K_0$. The constant here is actually total factor productivity so it should depend on both factors. Also generally for many diff. eq. constant will be function of some initial conditions I recommend you to have a look at further mathematics for economic analysis by Sydsæter et al chapter 5-7 $\endgroup$ – 1muflon1 Oct 2 '20 at 20:53
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If one reads the original article by Cobb and Douglas (1928), https://www.aeaweb.org/aer/top20/18.1.139-165.pdf ,

one will find at the end of page 152 that the authors stress that they took into account two properties that had been theoretically discussed in the past:

  1. That production exhibits constant returns to scale, meaning that doubling all inputs will double output.

  2. That both production inputs are necessary for production, so output should be zero when either one is zero.

To satisfy the 2nd property, they chose the multiplicative form. Given this, to satisfy the first property they had to make the sum of the exponents equal unity, so $a$, and $1-a$. We have $0<a<1$ so that output responds positively to each output.

The existence of the constant $A$ in $Q = AK^aL^{1-a}$ takes care of two things: any "units of measurement" issues but also, the average of any other forces that may contribute to output.

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Similarly to what 1muflon said, there's really nothing sacred or true about the Cobb-Douglas production function. It's just a simple function that has some desirable properties. (e.g. that marginal productivities are equal to average productivities.)

Things resembling "proofs of the Cobb-Douglas production function" that you might come across will be proofs that this function satisfies certain properties, or that it is the only function satisfying certain properties, not that it perfectly describes production in reality, which it does not.

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