Chapter 12.2, Problem 42E

Cobb-Douglas production function The output Q of an economic system subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function Q(L, K) = cL^aK^b, where a. b, and c are positive real numbers. When a + b = 1, the case is called constant returns to scale. Suppose a = $\frac{1}{3}$, b = $\frac{2}{3}$, and c = 40.

a.    Graph the output function using the window [0, 20] × [0, 20] × [0, 500].

b.    If L is held constant at L = 10, write the function that gives the dependence of Q on K.

c.    If K is held constant at K = 15, write the function that gives the dependence of Q on L.