Maximization with disposable income
Disposable Income $Y_D$ as a function of $Y_G$ and $Y_E$ is: 
$$Y_D = Y_G + (1-\alpha) Y_E$$
Given wage rate $w$, and $Y_E = wH$, the budget equation can be written as:
$$Y_D = Y_G + (1-\alpha)wH$$
Preferences of the individual over $Y_D$ and $H$ are represented by the utility function
$$U(Y_D, H) = \dfrac{Y_D}{(1+H)^2}$$
So the utility maximization problem is :
\begin{eqnarray*}\max_{Y_D, H} &amp;&amp; \ \ \dfrac{Y_D}{(1+H)^2} \\ \text{s.t} &amp;&amp;  \ \ Y_D = Y_G + (1-\alpha)wH \\ &amp;&amp;  \ \ H\geq 0 \end{eqnarray*}
Here is the picture to demonstrate optimal solution :





Solving the above problem we get the optimal labor supply as
\begin{eqnarray*} H^*(w, \alpha, Y_G) = \begin{cases} 1-\dfrac{2Y_G}{(1-\alpha)w} &amp; \text{if } (1-\alpha)w - 2Y_G &gt; 0  \\ 0 &amp; \text{if } (1-\alpha)w - 2Y_G \leq 0 \end{cases}  \end{eqnarray*}
So the optimal utility as a function of $w, \alpha, Y_G$ is as follows:
\begin{eqnarray*} U^*(w, \alpha, Y_G) = \begin{cases}  \dfrac{((1-\alpha)w)^2}{4((1-\alpha)w - Y_G)} &amp; \text{if } (1-\alpha)w - 2Y_G &gt; 0  \\ Y_G &amp; \text{if } (1-\alpha)w - 2Y_G \leq 0 \end{cases}  \end{eqnarray*}

Now the Govt. chooses $(Y_G, \alpha)$ by solving :
\begin{eqnarray*}\max_{Y_G, \alpha} &amp;&amp; \ \ U^*(w, \alpha, Y_G) \\ \text{s.t} &amp;&amp;  \ \ Y_G = \alpha wH^*(w, \alpha, Y_G)  \end{eqnarray*}


Solving it we get
$\alpha^* = 0$ and $Y_G^* = 0$