Consider a welfare system under which a single cash transfer (a guaranteed income) is given to every citizen. Then for each dollar the person earns, this payment is reduced by $\alpha$ dollars, where $0<\alpha<1$. Let $Y_G\geq 0$ be the guaranteed income. Then, if earned income is $Y_E$, the cash transfer is reduced by $\alpha Y_E$ dollars. So the net transfer is $T=Y_G-\alpha Y_E$. If $T>0$, you get a transfer. If $T<0$, you pay taxes. If $T=0$, you don't pay taxes or receive transfers.
Suppose that the wage rate is $w>0$ and so a person who works $H$ hours gets $Y_E=wH$. A typical individual has preferences over disposable income and hours of work represented by the utility function: $$U(Y_D,H)=\dfrac{Y_D}{(1+H)^2}$$ Imagine the government wants to choose its policy $(Y_G, α)$ in such a way that a typical citizen's utility is maximized, subject to the constraint that the net transfer is zero. Derive the optimal policy and carefully explain its properties.
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, $$a^*=2YG^*/w$$