Question
$0<a<1 $
$YG\ge 0$
$YE$ is positive
$T=YG-aYE$
$YD=YG+(1-a)YE$
Wage w is positive.
$YE=wH$
Utility function $$U(Y_D,H)=YD/(1+H)^2$$
Imagine that we need to choose $(YG, a)$ in such a way that utility function is maximized subject to the constraint that T=0. How can we derive the optimal $(YG, a)$.
My solution
$$L=(YD/(1+H)^2)-\lambda (YG-aYE)$$
$$L=((YG+(1-a)YE)/(1+H)^2)-\lambda (YG-aYE)$$
$$L=((YG+(1-a)wH)/(1+H)^2)-\lambda (YG-awH)$$
Derivative with respect to H
$$\frac{w(1+H)^2-2wH(1+H)}{(1+H)^4}=0$$
$$w(1+H)^2-2wH(1+H)=0$$
$$H^*=1/2$$
Then $$YG=aw/2$$
So, $$a^*=2YG^*/w$$
My solution is correct or not? I am not sure at all.