Maximizing Tax Revenue Given Supply and Demand!

Question

P^S=10+Q^S

P^D=100-2Q^D

If the government imposes a per-unit (excise) tax in this market, what is the maximum tax revenue that the government can collect?

Answer 1

Suppose the government impose a tax rate t dollar per unit in this market , after imposing tax

In the market equilibrium we alwasy have Pd=Ps+t, QD=QS=Q

Therefore we can plug in the demand and supply curve, we will have

Ps=10+Q (1)

Ps+t=100-2Q (2)

Plug (1) into (2), then we will have 10+Q+t=100-2Q , then t=90-3Q (3)

We know the tax revenue for the government is calculated by Tax Revenue=t*Q (4)

Plug in (3) into (4)

Tax Revenue=(90-3Q)*Q

Rearrange it we can have Tax Revenue=-3((Q-15)²-225)

Therefore the maximum tax revenue will be obtained when Q=15, Max(Tax Revenue)=675