Deriving the First-Order Condition for General Preferences Using the Chain Rule
To derive the first-order condition for a constrained choice problem with general preferences, differentiation is performed after using the substitution method. Once the constraint is substituted into the objective function, such as the utility function u(m_f^0 - τ, Q), both arguments of the function become dependent on the production quantity, Q. As a result, the chain rule is required when differentiating the objective function with respect to Q. The final step is to calculate this derivative, which yields the first-order condition for the optimal quantity. [4]
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