How Does the Envelope Theorem Apply to Nonlinear Functions Like f(x,r)?

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In summary, the conversation discusses the function f(x,r) = √x - rx, where x ≥ 0, and its maximum value function f*(r). The speaker is confused about how f*(r) can envelope the different (x,r) functions when f(x,r) is linear for a given x. They are looking for help in solving this problem.
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Peterw222
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Hi. I'm trying to solve the following problem; am confused.

Define the function f(x,r) = √x - rx, where x ≥ 0. Sketch both the function for several values of x, and the value function f*(r), which is the maximum value of f(x,r) for each given value of r. Describe how the function f*(r) is an envelope of the different (x,r) functions.

But f(x,r) is linear for given x - how can f*(r) envelope linear functions? Can anyone help?
 
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  • #2
Peterw222 said:
Hi. I'm trying to solve the following problem; am confused.

Define the function f(x,r) = √x - rx, where x ≥ 0. Sketch both the function for several values of x, and the value function f*(r), which is the maximum value of f(x,r) for each given value of r. Describe how the function f*(r) is an envelope of the different (x,r) functions.

But f(x,r) is linear for given x - how can f*(r) envelope linear functions? Can anyone help?

Welcome on MHB Peterw222!...

... the function...

$\displaystyle f(x,r) = \sqrt{x} - r\ x\ (1)$

... is linear in r and non linear in x... what You have to do is, given r, find the value x* that maximizes (1) and construct f*(r) = f(x*,r)...

Kind regards

$\chi$ $\sigma$
 

FAQ: How Does the Envelope Theorem Apply to Nonlinear Functions Like f(x,r)?

What is the Envelope Theorem?

The Envelope Theorem is a mathematical principle that states that the optimal value of a function can be found by taking the derivative of the function with respect to a parameter and setting it equal to zero. This allows for a simpler and more efficient way to find the optimal solution.

How is the Envelope Theorem used in economics?

In economics, the Envelope Theorem is used to find the optimal solution for a given problem. This can include finding the optimal level of production or consumption, or determining the optimal price for a product. It is also used in cost-benefit analysis to determine the optimal level of investment.

Can the Envelope Theorem be used for nonlinear functions?

Yes, the Envelope Theorem can be used for nonlinear functions. It is a general principle that applies to any differentiable function, regardless of its complexity. However, the calculations may be more challenging for nonlinear functions compared to linear ones.

What are the assumptions of the Envelope Theorem?

The Envelope Theorem assumes that the objective function is differentiable and that the constraints are continuously differentiable. It also assumes that the optimal solution is a unique maximum or minimum.

Are there any limitations to the Envelope Theorem?

While the Envelope Theorem is a useful tool for finding optimal solutions, it does have some limitations. It only applies to deterministic problems, meaning that there is a known relationship between inputs and outputs. It also assumes that the decision maker is rational and has perfect information about the problem.

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