Setting the Lagrangian for the first 50% of the population:
\(\mathcal{L}=ln(X)+ln(Y)+\lambda[P_x+P_Y-X P_X-Y P_Y]\)
F.O.C:
\(\frac{\partial \lambda}{\partial X} = 0= \frac{1}{X} -\lambda P_X\)
\(\frac{\partial \lambda}{\partial Y} = 0= \frac{1}{Y} -\lambda P_Y\)
\(\frac{P_X}{P_Y}=\frac{Y}{X}\)
Substituting in the budget constraint to find the individual demand function:
\(P_X+P_Y=P_X X+ P_X X\)
\(X=0.5+0.5\frac{P_Y}{P_X}\)
And:
\(Y=0.5+0.5\frac{P_X}{P_Y}\)
Setting the Lagrangian for the rest of the population:
\(\mathcal{L}=2*ln(X)+ln(Y)+\lambda[P_x+P_Y-X P_X-Y P_Y]\)
F.O.C:
\(\frac{\partial \lambda}{\partial X} = 0= \frac{2}{X} -\lambda P_X\)
\(\frac{\partial \lambda}{\partial Y} = 0= \frac{1}{Y} -\lambda P_Y\)
\(\frac{P_X}{P_Y}=\frac{2Y}{X}\)
Substituting in the budget constraint to find the individual demand function:
\(P_X+P_Y=P_X X+ \frac{P_X}{2} X\)
\(X= \frac{2}{3}+\frac{2}{3}\frac{P_Y}{P_X}\)
And:
\(P_X+P_Y=P_Y Y+ 2P_Y Y\)