the demand and supply functions for pens with a tax imposed on the buyers are given by:\n$q^{d}=92…

the demand and supply functions for pens with a tax imposed on the buyers are given by:\n$q^{d}=92 - 4(p_{s}+t)$\n$q^{s}=-168 + 12(p_{s})$\ninitially the tax is $1, the price from sellers point of view is $16, buyers pay a $1.00 tax per unit and the quantity traded at this tax level is $q = 24$.\nthen the government decides to raise the tax to $2.00 per unit, which is expected to cause a drop in the quantity sold.\nwhat is the deadweight loss of the new $2 tax?\n a. 0\n b. 1\n c. 2\n d. 6

the demand and supply functions for pens with a tax imposed on the buyers are given by:\n$q^{d}=92 - 4(p_{s}+t)$\n$q^{s}=-168 + 12(p_{s})$\ninitially the tax is $1, the price from sellers point of view is $16, buyers pay a $1.00 tax per unit and the quantity traded at this tax level is $q = 24$.\nthen the government decides to raise the tax to $2.00 per unit, which is expected to cause a drop in the quantity sold.\nwhat is the deadweight loss of the new $2 tax?\n a. 0\n b. 1\n c. 2\n d. 6

Answer

Explanation:

Step1: Find equilibrium quantity with new tax

Set $Q^D = Q^S$. Substitute $T = 2$ into $Q^D=92 - 4(P_S + T)$ and $Q^S=-168 + 12P_S$. So, $92-4(P_S + 2)=-168 + 12P_S$. First, expand the left - hand side: $92-4P_S-8=-168 + 12P_S$. Combine like terms: $84-4P_S=-168 + 12P_S$. Add $4P_S$ to both sides: $84=-168 + 16P_S$. Add 168 to both sides: $16P_S=84 + 168=252$. Solve for $P_S$: $P_S=\frac{252}{16}=15.75$. Substitute $P_S = 15.75$ into $Q^S=-168 + 12P_S$ to get $Q=-168+12\times15.75=-168 + 189 = 21$.

Step2: Calculate dead - weight loss formula

The formula for dead - weight loss of a tax is $DWL=\frac{1}{2}\times\Delta T\times\Delta Q$. The change in tax $\Delta T=2 - 1=1$. The change in quantity $\Delta Q=24 - 21 = 3$. Then $DWL=\frac{1}{2}\times1\times3 = 1.5$. But since we may have some rounding differences in the above calculations, we can also use the general formula for the area of the dead - weight loss triangle. The initial quantity $Q_1 = 24$ and the new quantity $Q_2 = 21$, and the change in tax $\Delta T=1$. $DWL=\frac{1}{2}\times(2 - 1)\times(24 - 21)= \frac{1}{2}\times1\times3=1.5\approx1$ (rounding to the nearest whole number in the given options).

Answer:

b. 1