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Identifying substitutes and complements
50 Part I: The Nature of Managerial Economics
Figure 3-2:
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Optimization: Profit maximization.
Putting It All Together: Optimization, Constraints, and the Lagrangian Function
Business situations are further complicated by constraints — perhaps the business has signed a contract to produce 1,000 units of the good daily, or the business has certain inputs, such as the factory size, that can’t be changed. Constraints limit the firm’s options. In these situations, your goal is to optimize a function subject to the limitations or constraints. For example, your firm wants to minimize the cost of producing the 1,000 units of output daily as specified by a contract it has with a customer.
The Lagrangian function is a technique that combines the function being optimized with functions describing the constraint or constraints into a single equation. Solving the Lagrangian function allows you to optimize the variable you choose, subject to the constraints you can’t change.
Identifying your objective (function)
The objective function is the function that you’re optimizing. The dependent variable in the objective function represents your goal — the variable you want to optimize. Examples of objective functions include the profit function to maximize profit, the cost function to minimize costs, and the utility function for consumers to maximize satisfaction (utility).
