Solve a CES production function maximization profit

Questions
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1

Hello All,

I’m trying to solve this simple CES production function profit maximization problem. These is the lagrangian and I need to find “Y”, “Xy” and “ny”, the other letters are just parameters.

L = Py Y - Px Xy - 
   Pn ny + \[Lambda] (Log[A] + v \[Beta] Log[Xy] + 
      v (1 - \[Beta]) Log[ny] - 
      0.5 \[Rho] v \[Beta] (1 - \[Beta]) Log[Xy] Log[Xy] - 
      0.5 \[Rho] v \[Beta] (1 - \[Beta]) Log[ny] Log[
        ny] + \[Rho] v \[Beta] (1 - \[Beta]) Log[Xy] Log[ny] - Y);
vars = {Y, Xy, ny, \[Lambda]};
Grid[Solve[Thread[D[L, #] & /@ vars == 0], vars], Alignment -> Left]

I attach the file so you can see that I got this no solution answer from the Mathematica software.

Solve::inex: "Solve was unable to solve the system with inexact coefficients 
or the system obtained by direct rationalization of inexact numbers present in 
the system. Since many of the methods used by Solve require exact input, 
providing Solve with an exact version of the system may help"
2

I don’t think Solve can handle this system. It has transcendental dependencies (use of both ny and Log[ny], for example). So you might have to restrict to numeric rather than symbolic parameters, and either use FindRoot or specify for Solve to work over the Reals (and it might still fail).