The Cobb-Douglas patient visit function for a clinic is given by f(x,y) = 1000x^(.7)y^(.3) where x represents numbers of doctors; y represents the numbers of nurses. If a doctor gets 42,000 dollars per quarter; a nurse, 9,000 dollars per quarter, and the total budget is 600,000 dollars.
a) Apply Lagrange multiplier to find x, y and λ for the maximum number of patient visits. b) What is the maximum number of visits? c) What is the meaning of Lagrange multiplier λ?
I'm getting that the maximum occurs at (10,20), λ = 61.557, and the maximum number of visits is 12,311. Could someone check me answer and tell me if I'm right?
Also could someone explain c to me. Thanks!
Your answer is correct! The maximum is at $(10,20)$.
$\lambda$ is the rate of change in the maximal value of the objective function as the constraint $\$600,000$ is relaxed.