Help with this microeconomics exercise
Given the production function
\begin{eqnarray*} F(L) = 10 \sqrt{L} - 2L \end{eqnarray*} 
marginal product is
\begin{eqnarray*} F'(L) = \frac{5}{ \sqrt{L}} - 2 \end{eqnarray*} 
Since there is no cost for the family for using its own labor and it cannot hire from outside, family will maximize its production subject to the constraint that it has 4 units of labor. 
\begin{eqnarray*} \max_{L\geq 0} &amp;&amp; 10 \sqrt{L} - 2L \\ \text{s.t.} &amp;&amp; L \leq 4 \end{eqnarray*} 
Given that $F'(L)$ is falling in $L$ and is positive at $L = 4$ $(F'(4) = 0.5)$, family will use all 4 units in optimum. The corresponding output is $10\sqrt{4} - 2(4) = 12$. 

With $10\%$ income tax, the objective of the firm will be to maximize total output net of tax which is equal to $0.9 F(L)$. Solution to this problem will stay the same as before because we are just multiplying the objective by a positive constant.

\begin{eqnarray*} \max_{L\geq 0} &amp;&amp; 0.9(10 \sqrt{L} - 2L) \\ \text{s.t.} &amp;&amp; L \leq 4 \end{eqnarray*} 

Therefore, the equilibrium output is still $10\sqrt{4} - 2(4) = 12$ out of which 1.2 is paid as taxes and 10.8 is consumed.

With new technology the problem is
\begin{eqnarray*} \max_{L\geq 0} &amp;&amp; 11 \sqrt{L} - L - 15\\ \text{s.t.} &amp;&amp; L \leq 4 \end{eqnarray*} 

The above problem again give $L = 4$ as the solution and the corresponding equilibrium output is $3$ which is way less than what the family had earlier. Therefore, it will not use this technology.