# Math Help - A question about Cobb-Douglas production function

1. ## A question about Cobb-Douglas production function

I have a problem of finding the solution to question related to the Cobb-Douglas production function. I don't know whether this is the right forum for me to ask the question. Nevertheless, the question is as follows:

For the Cobb-Douglas production function, $Q = AK^aL^b$, determine how the returns to scale depend on a and b. What special form does the function take in the case of constant returns to scale?

The answer given in the textbook is "Decreasing if a + b < 1, constant if a + b = 1, increasing if a + b > 1". And the answer to the second sub-question is "Q = AK^aL^(1-a) (A > 0, 0 < a < 1).

Can someone explain how I can come to the answers? Thank you very much XD

2. Without more information on K and L, and whether a and b are retricted to be positive, there is no way to answer this. For example, if $K= L= 1/2$ then $Q= A(1/2)^{a+b}$ this is decreasing for any a+ b< 0, increasing for any a+ b> 0. But for $K= L= 2$, it is just the reverse.

What are the restrictions on a, b, K, and L?

3. Originally Posted by HallsofIvy
Without more information on K and L, and whether a and b are retricted to be positive, there is no way to answer this. For example, if $K= L= 1/2$ then $Q= A(1/2)^{a+b}$ this is decreasing for any a+ b< 0, increasing for any a+ b> 0. But for $K= L= 2$, it is just the reverse.

What are the restrictions on a, b, K, and L?
Oh, sorry, I forgot to include the restriction on A, a, and b. The restriction is (A, a, b > 0). Sorry about that.

4. I can understand and interpret the answers given in the textbook. However, I do not know how to get to the answers I found that the Cobb-Douglas production function is homogenous of degree 2 by using the definition for homogenous function. However, I don't know how to proceed further...