ENVELOPE THEOREM Meaning and
Definition
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The envelope theorem is a concept in economics and mathematical optimization that relates to the behavior of a function when a parameter or constraint is varied. Specifically, it characterizes the derivative of a function with respect to a parameter, while holding another variable constant at its optimum value.
In simpler terms, the envelope theorem states that when optimizing a mathematical model, the optimal value of a function remains the same even if the factor being optimized changes slightly.
The "envelope" in the envelope theorem refers to the envelope curve, which represents the optimal values of a function as a parameter or constraint varies. This curve is formed by differentiating the objective function with respect to the parameter and then evaluating it at the optimum.
For example, suppose we have a production problem where a firm's profit function is optimized by choosing input levels of labor and capital. The envelope theorem allows us to determine how the optimal profit changes when the price of labor changes, while capital remains fixed at its optimum level.
The envelope theorem has important implications in various fields, including microeconomics, macroeconomics, finance, and operations research. It provides a useful tool for analyzing optimization problems and understanding the relationship between variables and optimization results.
Overall, the envelope theorem is a mathematical concept that enables us to analyze the behavior of a function as a parameter changes, while keeping another variable constant at its optimum value.
Etymology of ENVELOPE THEOREM
The word "envelope theorem" originated from mathematics and economics. Here is the breakdown of its etymology:
1. Envelope: In mathematics, an envelope refers to a curve that is tangent to a family of curves at each point and encompasses them in some way. It represents the boundary or outer limit of a set of curves.
2. Theorem: In mathematics, a theorem is a statement that has been proven or demonstrated to be true using rigorous logical reasoning. It is a fundamental concept in mathematics and other scientific disciplines.
Therefore, the term "envelope theorem" was coined to describe a mathematical theorem that involves the concept of envelopes. The theorem is used mainly in economics, particularly in the field of optimization, to analyze how small changes in parameters affect the optimal solution of a problem.
