Labour Market: simple macroeconomic modeling
To find demand for labor, solve the firm's profit maximization problem :
\begin{eqnarray*} \max_N &amp; \ P(2N - 0.5N^2) - WN\end{eqnarray*}
Differentiating the objective with respect to $N$ and setting it equal to 0 yields the demand function for labor :
\begin{eqnarray*} N^d = 2 - \frac{W}{P}\end{eqnarray*}
Corresponding supply of output is

\begin{eqnarray*} Y = f\left(2 - \frac{W}{P}\right) = \left(2 - \frac{W}{P}\right)\left(1 + \frac{W}{2P}\right) =  2 - \frac{1}{2}\left(\frac{W}{P}\right)^2\end{eqnarray*}

Given the supply of labor is
\begin{eqnarray*} N^s = \frac{W}{P}\end{eqnarray*}

Equilibrium in the labor market occurs where $N^d = N^s$. Solving it, we get equilibrium employment, real wage, output, profits as follows :
\begin{eqnarray*} N^* &amp; = &amp; 1 \\ \left(\frac{W}{P}\right)^* &amp; = &amp; 1 \\ Y^* &amp; = &amp; 1.5 \\ \pi^* &amp; = &amp; 0.5 \end{eqnarray*}