Question on Demand-Supply curves

Question

Assume that the market for cigarettes in a specific town has the given supply and demand curves: Q^S = P; Q^D = 50 − P, here the quantities are evaluated in thousands of units. Assume that the town council requires raising $300,000 in revenue and decides to do this by taxing the cigarette market. What must be the excise tax in order to increase the needed amount of money?

Answer 1

Let assume that the needed tax is $T. Then in equilibrium, P^D = P^S + T. This entails that 50 – Q = Q + T, or Q = 25 – 0.5T.  As the needed quantity is $300,000, we should have T*Q = 600. (Keep in mind that Q is measured in thousands of units). Therefore T (25 – 0.5T) = 600. By solving this equation we obtain two possible values for tax: T = $20 or T = $30.  Either one would produce $300,000 in tax revenues, although of course T = $20 would do so with much smaller deadweight loss.