WEW-054

Microeconomics Question from Walter E. Williams:

The Cobb-Douglas production function has the following form: $q = f(x,y) = Ax^a y^b$ : where $b = (1-a)$ Show that such a function is linearly homogeneous to degree one and state two other characteristics of such a production function. Also prove mathematically that with a production function such as a Cobb-Douglas that if factors are paid according to their marginal products the total product is "exhausted".

Answer

Homogenous to the degree one means the exponents when summed equal 1. The function has constant returns to scale. If each input costs its marginal product then the sum of the factor payments equals total revenue, so there are no economic profits. (Alternatively, if the sum of the exponents is greater than one, then the marginal product times the quantity of the input when summed over all inputs will exceed total revenue. This is an increasing returns to scale function.)

        $f(kx,ky) 	= A(kx)^a(ky)^b
		=Ak^ax^ak^by^b
		=Akx^ay^b
		=kf(x,y)$

Additional properties:

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