Question

Consider a market where supply and demand are given by ${\mathbf{Q}}_{\mathbf{x}}^{\mathbf{s}}\mathbf{=}\mathbf{-}\mathbf{16}\mathbf{+}{\mathbf{P}}_{\mathbf{x}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{Q}}_{\mathbf{x}}^{\mathbf{d}}\mathbf{=}\mathbf{92}\mathbf{-}\mathbf{2}{\mathbf{P}}_{\mathbf{x}}$.Suppose the government imposes a price floor of \(40, and agrees to purchase any and all units consumers do not buy at the floor price of $\mathbf{\backslash )}\mathbf{40}$ per unit.

a. Determine the cost to the government of buying firms’ unsold units.

b. Compute the lost social welfare (deadweight loss) that stems from the $\mathbf{\$}\mathbf{40}$price floor

Short answer

1. Quantity demanded is$12$ units
2. Dead weight loss is$\mathrm{\$}456$ units

Step-by-step solution

Given Information

$\begin{array}{l}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=-16+{\mathrm{P}}_{\mathrm{x}}\\ {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=92-2{\mathrm{P}}_{\mathrm{x}}\end{array}$

Finding the quantity demand and dead weight loss:

Given, the supply and demand functions are

$\begin{array}{l}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=-16+{\mathrm{P}}_{\mathrm{x}}\\ {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=92-2{\mathrm{P}}_{\mathrm{x}}\end{array}$

To determine the quality and demand supplied, we use the following steps -

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=-16+{\mathrm{P}}_{\mathrm{x}}\\ =-16+40\\ =24\end{array}$

Thus, the quantity supplied is 24 units

Again,

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=92-2{\mathrm{P}}_{\mathrm{x}}\\ =92-(2\times 40)\\ =92-80\\ =12\end{array}$

Thus, the quantity demanded is 12 units

$\mathrm{Excess}\mathrm{Units}\mathrm{Supplied}={\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}-{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}$

Substituting the given equation${\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=12\mathrm{and}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=24$

$\begin{array}{c}\mathrm{CST}={\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}-{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}\\ =24-12\\ =12\end{array}$

The excess unit supplied is$12$units.

By multiplying the excess unit supplied to the amount of price floor, we get

$12\times \mathrm{\$}40=\mathrm{\$}480$

The cost of buying firms unsold to the government is$\mathrm{\$}480$ units

To calculate the area of the triangle as follows-

$\begin{array}{c}\mathrm{DWL}\mathrm{Area}=\mathrm{Area}\mathrm{A}+\mathrm{Area}\mathrm{B}+\mathrm{Area}\mathrm{C}\\ =\frac{1}{2}\times (24-12)\times (40-36)\\ =\frac{1}{2}\times 12\times 4\\ =\frac{1}{2}\times 48\\ =24\end{array}$

By subtracting the cost to the government with the area of dead weight loss, we get

$\mathrm{\$}480-\mathrm{\$}24=\mathrm{\$}456$

Thus, the dead weight loss is $465units.