this is based off of Anthony and Brigg's Mathematics For Economics And Finance: Methods And Modeling.
it's exercise 1.5 on pg 11.
the question reads, "Find a formula for the amount of money the government obtains from taxing the commodity in the manner described in exercise 1.4. Determine this quantity explicitly when T=0.5."
i'll give you the required equations:
qS = 6p - 12 qD = 40 - 2p
the excise tax = pT = (13/2)+(3/4)T
the answer = 27T - (3/2)T^2
i have no clue how they came up with that answer. i got 27 + (9/2)T.
my process was to substitute the value of pT into qS.
i also tried putting pT-T into qS, which gets me 27 + (3/2)T, close but not exactly the answer.
also, i just followed the example in the book so i don't really have a conceptual understanding of the why, so can someone explain the logic behind inserting pT-T vs pT into qS?
here's the link for the book Mathematics for Economics and Finance: Methods and Modelling - Martin Anthony, Norman Biggs - Google Books
the question's on page 11.
Without tax $$ \begin{align} \text{Demand}\quad D:\quad & &q+2p&=40\\ \text{Supply}\quad S:\quad & &q-6p&=-12 \end{align} $$ that is the equilibrium is $$(p_e, q_e)=\left(\frac{13}{2},27\right)$$ With tax $T$ $$ \begin{align} \text{Demand}\quad D^T:\quad & &q^T+2p^T&=40\\ \text{Supply}\quad S^T:\quad & &q^T-6(p^T-T)&=-12 \end{align} $$ that is the equilibrium after tax is $$ (p^T_e,q^T_e)=\left(\frac{13}{2}+\frac{3}{4}T,\;27-\frac{3}{2}T\right)=\left(p_e+\frac{3}{4}T,\;q_e-\frac{3}{2}T\right) $$ We see that $p_e^T>p_e$ and $q^T_e<q_e$. So the government revenue is $$ GR=T\times q^T_e=T\times \left(27-\frac{3}{2}T\right)=27 T-\frac{3}{2}T^2 $$ and for $T=0.5$ we have $$ GR\big|_{T=0.5}=27 \times 0.5-\frac{3}{2}\times (0.5)^2=13.125 $$