9.11 More on Euler’s theorem Suppose that a production function f(x1, x2, ..., xn) is homogeneous of degree k. Euler’s theorem shows that P i xifi ¼ kf , and this fact can be used to show that the partial derivatives of f are homogeneous of degree k – 1.

a. Prove that Pn i¼1 Pn j¼1 xixj fij ¼ kðk $ 1Þf .

b. In the case of n ¼ 2 and k ¼ 1, what kind of restrictions does the result of part

(a) impose on the second-order partial derivative f12? How do your conclusions change when k > 1 or k < 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(x1, x2, ..., xn) ¼ Qn i¼1x ai i for ai + 0?