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

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.

• Does price/inverse demand have a specified unit, e.g. is $x_0$ a numeraire good? If not, you could normalize the price unit in such a way that $\lambda$ becomes one. May 15 at 8:23
• @Giskard Yes, $x_0$ is a numeraire good. Then why do we have $\lambda$ to be one when we have the numeraire good? May 15 at 11:23
• I don't know, I wrote a suggestion that works if it was not a numeraire. May 15 at 13:09

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 $$\lambda = 1$$.