Envelope Theorem

Definition

The envelope theorem is a fundamental concept in microeconomic theory that describes how changes in exogenous parameters affect the optimal choice of a decision variable. It provides a way to analyze how the optimal solution to an optimization problem varies with changes in the parameters of that problem.

5 Must Know Facts For Your Next Test

1. The envelope theorem is particularly useful in the analysis of cost minimization and profit maximization problems, where the decision variable is a function of exogenous parameters.
2. The envelope theorem allows us to determine how the optimal choice of a decision variable, such as the cost-minimizing input mix or the profit-maximizing output level, changes in response to changes in the exogenous parameters.
3. The theorem states that the derivative of the optimal value function with respect to an exogenous parameter is equal to the partial derivative of the objective function with respect to that parameter, evaluated at the optimal choice of the decision variable.
4. The envelope theorem can be used to derive important results in microeconomic theory, such as the Shephard's Lemma, which relates the cost-minimizing input demand to the firm's cost function.
5. The envelope theorem is a powerful tool for analyzing the comparative statics of optimization problems, as it allows us to determine the direction and magnitude of the change in the optimal solution without having to solve the optimization problem again.

Review Questions

• Explain how the envelope theorem can be used to analyze the cost-minimizing input mix of a firm.
  • The envelope theorem can be used to analyze how changes in exogenous parameters, such as input prices, affect the cost-minimizing input mix of a firm. Specifically, the theorem states that the derivative of the firm's cost function with respect to an input price is equal to the optimal quantity of that input. This relationship, known as Shephard's Lemma, allows the firm to determine how its cost-minimizing input demands will change in response to changes in input prices, without having to solve the full cost minimization problem again.
• Describe how the envelope theorem can be used to analyze the profit-maximizing output level of a firm.
  • The envelope theorem can also be used to analyze how changes in exogenous parameters, such as output price or input prices, affect the profit-maximizing output level of a firm. In this case, the theorem states that the derivative of the firm's profit function with respect to an exogenous parameter is equal to the optimal value of the corresponding choice variable. For example, the derivative of the profit function with respect to output price is equal to the profit-maximizing output level. This relationship allows the firm to determine how its profit-maximizing output will change in response to changes in output price or input prices, without having to solve the full profit maximization problem again.
• Explain how the envelope theorem can be used to derive important results in microeconomic theory, such as Shephard's Lemma.
  • The envelope theorem is a powerful tool for deriving important results in microeconomic theory, such as Shephard's Lemma. Shephard's Lemma states that the derivative of a firm's cost function with respect to an input price is equal to the cost-minimizing quantity of that input. This result can be derived directly from the envelope theorem, which states that the derivative of the optimal value function (in this case, the cost function) with respect to an exogenous parameter (in this case, the input price) is equal to the partial derivative of the objective function (the cost function) with respect to that parameter, evaluated at the optimal choice of the decision variable (the cost-minimizing input mix). This relationship between the firm's cost function and its input demands is a fundamental result in microeconomic theory, and the envelope theorem provides a powerful way to derive and understand it.

Related terms

Optimization Problem: An optimization problem is the mathematical formulation of a decision-making situation where the goal is to find the best solution from a set of feasible alternatives, given certain constraints.

Lagrange Multiplier: A Lagrange multiplier is a variable introduced in the optimization of a function subject to equality constraints. It represents the marginal change in the objective function due to a marginal change in the constraint.

Comparative Statics:

Comparative statics is the analysis of how the equilibrium of a system changes when one of the parameters of the system is changed, while holding all other parameters constant.