The Envelope Theorem is a fundamental result in optimization theory with wide applications in economics. It provides a powerful tool for analyzing how optimal values change when parameters change. Consider an optimization problem where we choose x to maximize function f of x and parameter a. As parameter a changes, both the shape of the function and the optimal choice of x change.
Let's formalize the envelope theorem mathematically. We have an optimization problem where we choose x to maximize function f of x and parameter a. The optimal choice x star depends on parameter a. The value function V of a gives the maximum value of f for each parameter value. The envelope theorem states that the derivative of the value function with respect to parameter a equals the partial derivative of f with respect to a, evaluated at the optimal point. The envelope curve connects all optimal points as parameter a changes.
The intuition behind the envelope theorem is crucial for understanding its power. At the optimal point x star, the objective function is locally flat with respect to small changes in x. This means the first-order condition is satisfied - the derivative with respect to x equals zero. When parameter a changes, there are two potential effects: a direct effect from a appearing in the function, and an indirect effect from the optimal choice x star changing. However, because the function is flat at the optimum, the indirect effect is zero at the margin. Only the direct effect matters for the rate of change of the optimal value.
The envelope theorem has numerous important applications in economics. In cost minimization, Shephard's lemma states that the derivative of the minimum cost function with respect to an input price equals the optimal quantity of that input used. In profit maximization, Hotelling's lemma shows that the derivative of the maximum profit function with respect to an output price equals the optimal quantity of that output produced. In consumer theory, Roy's identity relates the derivative of the indirect utility function with respect to a good's price to the optimal quantity of that good consumed. These applications demonstrate the theorem's power in deriving comparative statics results without explicitly solving for how optimal choices change.
In conclusion, the envelope theorem is a powerful and elegant tool in economics that greatly simplifies comparative statics analysis. Instead of solving complex systems to determine how optimal choices change with parameters, the theorem provides direct relationships between parameters and optimal values. It forms the theoretical foundation for many important results in microeconomics, including Shephard's lemma, Hotelling's lemma, and Roy's identity. The theorem demonstrates the beauty and power of optimization theory in economics, showing how mathematical insights can provide deep economic understanding with remarkable simplicity.