Background and Trivial Proof by "Example"
The way the Rate of Return for the Cobb Douglas production function is proven in the new definition of my textbook is as follows.
The Cobb Douglas function definition
$ p = AL^aK^b$ $where,$ $(A,a,b)$ $are$ $constant$.
Consider multiplying the inputs Labor $L$ and Capital $K$ by any factor $m$.
$A(mL)^a(mK)^b$
$Am^aL^am^bK^b$
$m^am^bAL^aK^b$
$m^{(a+b)}AL^aK^b$
$m^{(a+b)}p$
From this you can see that the rate of change is dependent on the sum of $a+b$ and it is easy to show the properties of returns to scale as they simply follow the rules of the exponential function. The important properties in terms of economics is that for any positive change in m (means m != 1) the rate of change increases if $a+b>1$ decreases if $a+b<1$ and is equal to the change in m if $a+b=1$
Problem Statement
The previous method ignores defining mathematically Returns to Scale. Furthermore, in an older version of my text book it uses some calculus to prove that Returns to Scale equals $a+b$ … it just sort of skips some steps after the basic differentiation so I can't understand how they arrive at that answer.
Attempted Solution
Returns to Scale seems to be the change in the production function. If this is correct it should be equal to the derivative of the production function. Since we have two variables K and L we need to take the partial derivatives and add them together.
$R =\frac{∂P}{∂K} + \frac{∂P}{∂L}$
$R = aAL^{a-1}K^b + bAL^aK^{b-1}$
$R = aAL^aL^{-1}K^b + bAL^aK^bK^{-1}$
$R = AL^aK^b(aK^{-1} + bL^{-1})$
This is where the old edition textbook page I saw did some magic I'm not 100% sure but I think it said divide by p and evaluate at 1 to get $R = a + b$.
Notes
I'm not sure how to simplify it any further and ending up with the original function in the derivative confuses me even more. I haven't done partial derivatives in forever though so there may be something I'm just missing about adding partial derivatives.
I'm also confused because from the background proof it would seem that the Returns to Scale are $m^{a+b}$ not a+b.
Best Answer
Hint: In the case of the Cobb-Douglas function the return of scale is the sum of the elasticities of the input factors:
$$R=\frac{\partial P}{\partial L}\cdot \frac{L}{P}+\frac{\partial P}{\partial K}\cdot \frac{K}{P}$$
$$=a\cdot A\cdot L^{a-1}K^b\cdot \frac{L}{AL^{a}K^b}+b\cdot A\cdot L^{a}K^{b-1}\cdot \frac{K}{AL^{a}K^b}$$
I think you can finish.