9.11 More on Euler's theorem Suppose that a production function f(x1,...,x) is homogeneous of degree k. Euler's theorem shows that fkf, and this fact can be used to show that the partial derivatives of f are homogeneous of degree k-1.

a. Prove that *;*;f = k(k-1)f.

b. In the case of n = 2 and k = 1, what kind of restrictions does the result of part

(a) impose on the second-order partial derivative fi? How do your conclusions change when k > 1 ork < 1

c. How would the results of part

(b) be generalized to a production function with any number of inputs?

d. What are the implications of this problem for the parameters of the multivariable Cobb- Douglas production function f(x,,,x) = II for

a, 0?