I maximize firm’s expected profit with taking into account the quadratic cost of adjusting its nominal price according to the Ireland (1997) and Dib (2003). With the exception that constraints have changed. Can I change the constraints? This change of constraints to this form is right?

In early Lagrange equation, the expression in brackets after Lagrange multipliers is equal to the demand for good j as a function of its relative price and final output (yellow braces) that the first sentence is firm’s output in the beginning equation of the file. This phrase was written in terms of yjt that I have written next Lagrange equation according to pjt and I’ve put the firm’s output instead of yjt.
(I have specified important points with yellow and red)

You never really specified what the “element of” ,i.e.

in LaTeX means. What I infer from your document and your reply is that you still use the same constraint, but written differently by simply transforming the original equation using equivalence operations. Therefore, the two Lagrangians are perfectly equivalent (unless you divided by 0 in the course of doing this, which should by impossible in this context) and the approach is valid (although it is not clear that the rewriting will make things easier later on)