In the optimization exercises, the envelope theorem states :
Consider a utility maximization problem given the budget constraint:
Suppose I want to know how the maximum utility changes when price of good 1 (say) changes. I can calculate the optimum values of good 1 and good 2. Then i calculate the utility at these optimum. Then I can differentiate with respect to price of good 1.
Another way is i can differentiate the Lagrange with respect to price of good 1.
Both ways, I get the change in Utility with respect to change in price.
However, what is the intuition for this. Why differentiating the Lagrange with respect to price gives me the changes in utility when price changes.