Question

Suppose demand and supply are given by ${\mathbf{Q}}_{\mathbf{x}}^{\mathbf{d}}\mathbf{=}\mathbf{14}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{P}}_{\mathbf{x}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{Q}}_{\mathbf{x}}^{\mathbf{s}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}}{\mathbf{P}}_{\mathbf{x}}\mathbf{-}\mathbf{1}$

a. Determine the equilibrium price and quantity. Show the equilibrium graphically.

b. Suppose a$\mathbf{\backslash (}\mathbf{12}$excise tax is imposed on the good. Determine the new equilibrium price and quantity.

c. How much tax revenue does the government earn with the$\mathbf{\backslash )}\mathbf{12}$tax?

Short answer

1. Price and quantity demanded is $20,4$ units,
2. The equilibrium quantity is$4$units,
3. Tax revenue that the government earn$\$24$.

Step-by-step solution

Given information:

$\begin{array}{l}{\mathrm{Q}}_{\mathrm{X}}^{\mathrm{d}}=14-\frac{1}{2}{\mathrm{P}}_{\mathrm{X}}\\ {\mathrm{Q}}_{\mathrm{X}}^{\mathrm{S}}=\frac{1}{4}{\mathrm{P}}_{\mathrm{X}}-1\end{array}$

Determining equilibrium price and quantity

a.

$\begin{array}{l}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=14-\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}\\ {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=\frac{1}{4}{\mathrm{P}}_{\mathrm{x}}-1\end{array}$

${\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}={\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}$

Converting to$\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}\mathrm{to}0.5{\mathrm{P}}_{\mathrm{x}}\mathrm{and}\frac{1}{4}{\mathrm{P}}_{\mathrm{x}}-1\mathrm{to}0.25{\mathrm{P}}_{\mathrm{x}}-1$. Then, substituting the equation we get,

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}={\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}\\ 14-0.5{\mathrm{P}}_{\mathrm{x}}=0.25{\mathrm{P}}_{\mathrm{x}}-1\\ 0.75{\mathrm{P}}_{\mathrm{x}}=15\\ {\mathrm{P}}_{\mathrm{x}}=20\end{array}$

Thus, the equilibrium price is $ $20$.

To determine the equilibrium quantity

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=14-0.5{\mathrm{P}}_{\mathrm{x}}\\ =14-0.5\times 20\\ =14-10\\ =4\end{array}$

The equilibrium quantity is $4$ units

The graph

b.

The graph as shown in the equilibrium as

Determining tax revenue

c.

To calculate the tax revenue,

$\mathrm{TaxRate}=\mathrm{\$}12\mathrm{and}{\mathrm{Q}}^{\mathrm{d}}=2$

Substitute the given equations are

$\begin{array}{c}\mathrm{Tax}\mathrm{Revenue}=\mathrm{TaxRate}\times {\mathrm{Q}}^{\mathrm{d}}\\ =\mathrm{\$}12\times 2\\ =\mathrm{\$}24\end{array}$

Thus, the tax revenue that the government earns is $24.