Suppose now the inverse demand function is $P(Q)=-Q$ and the cost per unit of output is $z.$

$($a$)$ Find the profit maximizing output, $Q^{**},$ and the associated monopoly profit, $(Q^{**})$

$($b$)$ Find $\frac{\text{d}\text{e}\text{l}\text{d}}{\text{d}\text{e}\text{l}\text{d}\text{z}}$ to illustrate how monopoly profit reacts to a change in cost per unit, $z$

$($c$)$ Suppose the government wants the monopoly to produce $a-z$ units of output. What subsidy per unit of output, $s,$ must the government provide to elicit $Q^{**}=-z?$