Cost Minimization & Profit Maximization
The Lagrangian setup behind firm behavior, first-order conditions with economic meaning.
Definition
The firm's formal problems: choose inputs to minimize the cost of a target output, or choose output/inputs to maximize profit, both solved with first-order conditions.
Key equation
Profit max: pĀ·MPL = w, pĀ·MPK = r (pay each factor its marginal revenue product)
The intuition
The Lagrangian turns 'do the best you can under a constraint' into equations: at the optimum, the last dollar spent on any input buys the same output everywhere, and each factor is hired until its marginal value equals its price.
Exam tip
Cost minimization is nested inside profit maximization, solve the cost function first and MC = dTC/dQ falls out for free.
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