# Maximizing total tax revenue with function $Q_s=-8+P$ and $Q_d=\frac{80}{3}-\frac{1}{3}P$

The supply and demand equations of a good are given by

$Q_s=-8+P$

$Q_d=\frac{80}{3}-\frac{1}{3}P$

$P$ is measured in dollars. Suppose the government decides to impose a constant per unit tax of $t$ on the supplier.

Find the equilibrium quantity in terms of $t$ Using the expression found in part 1 find the value of $t$ that maximizes the governments total tax revenue. (Make sure the second order condition is satisfied)

I know equilibrium is when Qs=Qd but I don't know how to implement the $t$. Do I just add it the end of $Q_s$ making it $Q_s= (-8+P)+t$ ? If so how do I solve for the equilibrium?

The price supply-function for one unit, without tax, is $P=Q^s+8$. The tax is a unit tax in the amount of $\$t$. Thus the unit price increases by t.$P^s+t=Q^S+8+t$And the demand function is still$Q^d=(80/3) - (1/3)P$Solving for p$(1/3)P=80/3-Q^d\quad | \cdot 3P^d=80-3\cdot Q^d$In equilibrium it has to be$P^s+t=P^dQ+8+t=80-3\cdot Q \quad | +3Q4Q+8+t=80 \quad | \quad -84Q+t=72 \quad | -t4Q=72-t \quad | :4Q=18-\frac{t}{4} $As you can see from the graph below the governments total tax revenue is the green area. It is$T(t)=Q(t)\cdot t=18t-\frac{t^2}{4}\$.