Lagrangian with log and summation

In summary, the conversation discusses a microeconomics problem and the correctness of the first-order conditions (FOCs) for a utility maximization problem with a budget constraint. The FOCs and the use of a Lagrange multiplier are mentioned, along with a suggestion to double-check notation and include the inequality sign in the constraint.
  • #1
Kesyt3
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This is a microeconomics problem that I am trying to solve. I am uncertain whether my FOCs are correct. Thank you.

The objective function: ui(x1i, x2i….xLi) = Σllog[xli];
The constraint: ΣLl=1p1xl ≤ w

L: Σllog[xli] + λ (w - ΣLl=1p1xl)

FOCs are:
L1 = 1/x1 – λ(w-p1) =0
L2 = 1/x2 – λ(w-p2) =0
:
LLi = 1/xLi - λ(w-pLi) =0
Lλ = w - ΣLl=1p1xl = 0
 
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  • #2


Hi there,

Thank you for sharing your objective function and constraint. Your FOCs look correct to me. They represent the first-order conditions for a utility maximization problem with a budget constraint. The Lagrange multiplier, λ, is used to incorporate the constraint into the optimization problem.

However, I would recommend double-checking your notation. In your constraint, the summation should be over i instead of l, since the constraint applies to all goods (x) for a given individual (i).

Also, make sure to include the inequality sign (≤) in your constraint, as it represents a budget constraint rather than an equality constraint.

Keep up the good work and happy problem solving!
 

FAQ: Lagrangian with log and summation

What is a Lagrangian with log and summation?

A Lagrangian with log and summation is a mathematical function used in physics and engineering to calculate the dynamics of a system. It combines the use of logarithmic and summation functions to determine the equations of motion for a system.

How is a Lagrangian with log and summation used in physics?

In physics, a Lagrangian with log and summation is used to determine the equations of motion for a system by considering the different forces and constraints acting on the system. It takes into account the total kinetic and potential energy of the system and uses the principle of least action to find the most probable path of the system.

What is the principle of least action?

The principle of least action states that a physical system will follow the path that minimizes the action, which is the integral of the Lagrangian function over time. This principle is used in the Lagrangian with log and summation to find the equations of motion for a system.

How does the logarithmic function affect the Lagrangian?

The logarithmic function in the Lagrangian accounts for systems that have energy dissipation or damping, such as a pendulum swinging through air resistance. It helps to model the system more accurately and provides a more realistic representation of the system's dynamics.

Are there any limitations to using a Lagrangian with log and summation?

While the Lagrangian with log and summation is a powerful tool for analyzing the dynamics of a system, it does have some limitations. It may not be suitable for systems with highly nonlinear behavior or complex constraints. Additionally, it may be difficult to apply to systems with a large number of degrees of freedom.

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