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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?

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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

$(1/3)P=80/3-Q^d\quad | \cdot 3$

$P^d=80-3\cdot Q^d$

In equilibrium it has to be $P^s+t=P^d$

$Q+8+t=80-3\cdot Q \quad | +3Q$

$4Q+8+t=80 \quad | \quad -8$

$4Q+t=72 \quad | -t$

$4Q=72-t \quad | :4$

$Q=18-\frac{t}{4} $

As you can see from the graph below the governments total tax revenue is the green area. It is $T(t)=Q(t)\cdot t=18t-\frac{t^2}{4}$.

To maximize T(t) w.r.t t you have to differentiate T(t) and set it equal to zero. At the end you have to solve the equation.

enter image description here

Here is t=8. It it not the value, which maximizes the governments total tax revenue. The grey area is the producer surplus. The red area is the consumer surplus (not complete in the picture). The blue line is the supply curve without taxes. The red line is the supply curve with taxes. The green line is the demand curve.

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