Why can the Lagrangian Multiplier be dropped in the inverse demand function?

Question

I'm deriving the Antras and Helpman (2004) paper. The model assumes a nested CES utility function $$ U = x_0 + \frac{1}{\mu} \sum_{j=1}^{J} X_j^\mu $$ where $X_j = \left[ \int x_j(i)^\alpha \,di \right]^{1/\alpha}$.

The paper straightforwardly shows that the inverse demand function for each variety i in sector j is $$ p_j(i) = X^{\mu-\alpha}x_j(i)^{\alpha-1} $$

I know this can be obtained by setting up the Lagrangian function and finding the first-order condition with respect to $x_j(i)$. But why here we can drop the Lagrangian multiplier $\lambda$? What I actually found is $$ p_j(i) = X^{\mu-\alpha}x_j(i)^{\alpha-1} (1/\lambda) $$ I suspect in this case we can normalise $\lambda$ to 1 due to some implicit assumption that is not clearly stated in the paper. But I never heard that we could easily drop or normalise the Lagrangian multiplier in any of my previous economics and maths class.

Answer 1

The Lagrangian is given by: $$ L = x_0 + \frac{1}{\mu} \sum_{j = 1}^J X_j^\mu - \lambda(x_0 + \sum_j \int_i p_j(i) x_j(i) di - m). $$ The first order condition (for an interior solution) with respect to $x_0$ gives: $$ 1 - \lambda = 0 $$ So $\lambda = 1$.