Question

The supply curve for product X is given by ${\mathbf{Q}}_{\mathbf{x}}^{\mathbf{s}}\mathbf{=}\mathbf{-}\mathbf{520}\mathbf{+}\mathbf{20}{\mathbf{P}}_{\mathbf{x}}$.

1. Find the inverse supply curve.
2. How much surplus do producers receive when ${\mathbf{Q}}_{\mathbf{x}}\mathbf{=}\mathbf{400}$ ? When ${\mathbf{Q}}_{\mathbf{x}}\mathbf{=}\mathbf{1200}$?

Short answer

1. The inverse supply curve is ${\mathrm{P}}_{\mathrm{x}}=\frac{1}{20}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}+26$.
2. The surplus that producers receive if ${\mathrm{Q}}_{\mathrm{x}}=400$is $$46$ , and the surplus that producers receive if $Q_x=1,200$is $role="math" localid="1655270494630" $86$ .

Step-by-step solution

Given information 

$\begin{array}{l}{\mathrm{Q}}_{\mathrm{X}}^{\mathrm{S}}=-520+20{\mathrm{P}}_{\mathrm{X}}\\ {\mathrm{Q}}_{\mathrm{X}}=1,200,{\mathrm{Q}}_{\mathrm{X}}=400\end{array}$

Finding the inverse supply curve

a.

To find the inverse supply curve, we need the representation of a linear supply function:

${\text{Q}}_{\text{X}}^{\text{S}}={\mathrm{\beta }}_0+{\mathrm{\beta }}_{\text{X}}{\text{P}}_{\text{X}}+{\mathrm{\beta }}_{\text{y}}{\text{P}}_{\text{y}}+{\mathrm{\beta }}_{\text{W}}\text{W}+{\mathrm{\beta }}_{\text{H}}\text{H}$.

The supply equation for the product is

$\begin{array}{c}{\text{Q}}_{\text{X}}^{\text{S}}=-520+20{\text{P}}_{\text{X}}{\text{,P}}_{\text{X}}\\ =46\end{array}$

Thus, the inverse supply function is ${\mathrm{P}}_{\mathrm{X}}=\frac{{\text{Q}}_{\text{X}}^{\text{S}}+520}{20}$.

Finding the surplus that the producers receive

b.

When $Q_x=400$,the producer’s surplus is -

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=-520+20{\mathrm{P}}_{\mathrm{x}}\\ 400=-520+20{\mathrm{P}}_{\mathrm{x}}\\ 520+400=20{\mathrm{P}}_{\mathrm{x}}\\ {\mathrm{P}}_{\mathrm{x}}=\frac{920}{20}\\ {\mathrm{P}}_{\mathrm{X}}=46\end{array}$

The surplus the producers receive if ${\mathrm{Q}}_{\mathrm{x}}=400$ is $ $46$.

When ${\mathrm{Q}}_{\mathrm{x}}=1200$,the producer’s surplus is -

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=-520+20{\mathrm{P}}_{\mathrm{x}}\\ 1200=-520+20{\mathrm{P}}_{\mathrm{x}}\\ 520+1200=20{\mathrm{P}}_{\mathrm{x}}\\ {\mathrm{P}}_{\mathrm{x}}=\frac{1720}{20}\\ {\mathrm{P}}_{\mathrm{X}}=86\end{array}$

Thus, the surplus that the producers receive if ${\mathrm{Q}}_{\mathrm{x}}=1,200$ is $ $86$.