The Cobb-Douglas production function takes the form Q= AKa L1-a (a=alpha) where Q is the amount of output, K is the amount of capital input, and L is the amount of labor input. Suppose that a firm faces a linear cost-of-inputs function C= wL + rK where C is the cost of inputs, w is the wage rate, and r is the rental rate on capital
(a) Set up a Lagrangian function reflecting the constrained optimization problem of obtaining the most output given a budget C to spend on inputs. Solve this for the optimal levels of capital and labor (Answer: K= (C. alpha)/r , L= C .(1-alpha)/ w )
(b) Set up a Lagrangian function reflecting the constrained optimization problem of spending the least amount on inputs given that the level of output must equal the amount Q. Solve this for the optimal levels of capital and labor. (Answer: K= Q/A .(alpha/ 1-alpha) ^ 1-alpha . (w/r) ^ 1-alpha , L = Q/A . (alpha/ 1-alpha) ^ -alpha . (w/r)^ - alpha