Lagrangian with log and summation

[Kesyt3]

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

 

[jvicens]

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!