Envelope Theorem in Optimization Theory
In optimization theory, the Envelope Theorem simplifies analysis concerning how an optimal value of a function responds to changes in external parameters.
Throughout economics, it’s useful in understanding how changes in prices, income, or other parameters affect maximum utility or profit, without having to recalculate an entire optimization problem.
Let’s consider an optimization problem where an individual chooses an exogenous variable x to maximize a function L(x, p). p is a parameter, (think price) and the optimal value function is V(p) = maxₓL(x,p)
In this case, the Envelope Theorem states that the derivative of the optimal value function with respect to the parameter p equals the partial derivative of the objective function with respect to p, cumulatively evaluated at the optimal choice x*(p).
Envelope Theorem in Optimization Theory
This means that if we’re analyzing small changes in p(price), we can focus on the direct effect of p on the function while maintaining the decision variable at its optimal level, without having to consider how x* changes along with p. This makes comparative statics analysis much easier.
As an applied example, in consumer theory, we can use the Envelope Theorem to find the effect on maximum utility if we’re trying to figure out how a price change affects utility at the optimal consumption bundle, all without re-solving the utility maximization problem.
Hope that helps, and leave any other questions in the comments! Or, Check out my other articles on economics here (useful for studying, or just learning).