Lagrange multipliers are partial case of the so-called KKT conditions (Karush–Kuhn–Tucker conditions), and without the later all economic theory, and some finance applications as well, are gonna be in trouble in finite or infinite dimensional spaces.

The the Lagrange multipliers LG originally introduced in the Boltzmann-Einstein model to derive the Gaussian distribution are just a classical mathematical trick to compensate for the lack of true definition of probability in unified 4D space.

The derivation of the Boltzmann distribution for the energy distribution of identical but distinguishable classical particles can be obtained via a mathematical approach [1] or equivalently via a statistical approach [2,3] where the Lagrange multipliers can be completely ignored.

However, there are still many people who claim that

i- without the Lagrange multipliers all economic theory, and some finance applications as well, are gonna be in trouble in finite or infinite dimensional spaces.

ii- while the use of Lagrange multipliers may not be the only way to derive the Boltzmann distribution, it is a well-established and useful technique that should not be dismissed as a mere "trick" .

iii- without the L.G. all economic theory, and some finance applications as well, are gonna be in trouble in finite or infinite dimensional spaces.

iv-LG are used in various fields, including physics, economics, and optimization. They are used to optimize a function subject to a set of constraints, by introducing additional parameters (the Lagrange multipliers) that allow the constraints to be incorporated into the objective function.

In the context of statistical mechanics, Lagrange multipliers are used to enforce the constraints on the total energy, volume, and number of particles in a system, while maximizing the entropy.

While it's true that the use of Lagrange multipliers is a mathematical technique, it's not just a "trick" that can be ignored. The Lagrange multipliers are necessary to incorporate the constraints into the optimization problem and obtain the correct solution. Without the Lagrange multipliers, the constraints would not be taken into account, and the resulting distribution would not accurately reflect the physical behavior of the system.

We assume that the fears or claims i-iv will not happen because LG constraints will be incorporated in adequate statistical theory.

In brief, Lagrange multipliers is just a classic mathematical trick that we can do without.

A detailed answers to this question and other related questions such as the numerical statistical solution of double and triple integration as well as the time dependent PDE are all explained in references[2,3] where the modern definition of probability in the transition matrix B is an interconnected thing of the three topics. Imagination is the first important common factor in mathematics, physics and especially in the probability of transition in unitary 4D space is incorporated in B-matrix statistical chains to numerically solve single, double, and triple (Hypercube) integrals as well as time dependent PDE[ 2.3].

Ref:

1-The Boltzmann factor: a simplified derivation

Rainer Muller

2-I.M.Abbas, How Nature Works in Four-Dimensional Space: The Untold Complex Story, Researchgate, May 2023.

3-I.M.Abbas, How Nature Works in Four-Dimensional Space: The Untold Complex Story, IJISRT review, May 2023.