You have just emerged from medical school with a debt service burden of $25,000 per year, and have set up practice. You have to decide how hard to work . For each hour of work, you expect to earn 50 dollars ( after subtracting expenses of maintaining your office, taxes, etc.).
(a) What is your budget constraint linking I and H?
(b) Find your optimal number of hours of work.
I got the budget constraint as I= 50H - 25,000
After forming the Lagrangian, I'm getting I=75,000 and H=2000.
But for the second order derivatives, I'm getting the second order derivative with respect to I as 0 and that of H as 2I. Thus, the second order derivatives are positive which should be negative for a maximum. Also, the function is neither concave nor convex using the Hessian principal minors method.
I think there is no solution because the budget line is upward sloping and will intersect the indifference curve which leaves room for improvement.