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Math Help - A question about Cobb-Douglas production function

  1. #1
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    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
    Last edited by Real9999; August 16th 2010 at 02:55 AM.
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  2. #2
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    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?
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    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.
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  4. #4
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    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...
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