I know the answer to the first one is "e" and the answer to the second one is "a "but I have no idea why please explain how to do these two problems. I am very confused with this new method. Thank you!
The “Cobb-Douglas” productivity function for a factory is q = 100x0.8y0.2, where x is the number of workers, y is the number of machines, and q is the number of items the factor produces a year. Annual operating costs amount to $40,000 per worker and $2,000 per machine. This year the annual operating budget of the factory is $500,000. In setting up the problem of maximizing q as a constrained optimization problem and solving it using the Lagrange multiplier method, which of the following is WRONG?
(a) The function to be maximized is
q = 100x^0.8y^0.2.
(b) A constraint is 40, 000x + 2, 000y = 500, 000.
(c) The Lagrange function is L(x, y) = 100x^0.8y^0.2 − λ(40, 000x + 2, 000y − 500, 000).
(d) An equation to be satisfied at the optimal (x, y) is 80x−0.2y0.2 − 40, 000λ = 0.
(e) An equation to be satisfied at the optimal (x, y) is 80x−0.2y0.2 = 0.
In the previous question, what are the number of workers and the number of machines that maximize q? (Use the Lagrange multiplier method.)
(a) 10 workers and 50 machines
(b) 50 workers and 10 machines
(c) 40 workers and 2 machines
(d) 80 workers and 2 machines
(e) None of the above.