I understand that there are two common applications of Lagrangian multipliers in consumer theory: the utility maximization problem (UMP) ans the expenditure minimization problem (EMP).
These seem like the same problem to me but with different parameters being the focus. Why does my professor emphasize the importance of duality between these theories? I don't understand why both are needed.
To what extent are the UMP and EMP different economic ideas/problems as opposed to just simply being a different way of mathematically stating the same thing?