Question

Suppose the supply function for product $\mathbf{X}$is given by${\mathbf{Q}}_{\mathbf{x}}^{\mathbf{s}}\mathbf{=}\mathbf{-}\mathbf{30}\mathbf{+}\mathbf{2}{\mathbf{P}}_{\mathbf{x}}\mathbf{-}\mathbf{4}{\mathbf{P}}_{\mathbf{z}}$

a. How much of product $\mathbf{X}$is produced when${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash (}\mathbf{600}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{P}}_{\mathbf{z}}\mathbf{=}\mathbf{\backslash )}\mathbf{60}$?

b. How much of product$\mathbf{X}$is produced when${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash (}\mathbf{80}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{P}}_{\mathbf{z}}\mathbf{=}\mathbf{\backslash )}\mathbf{60}$?

c. Suppose${\mathbf{P}}_{\mathbf{z}}\mathbf{=}\mathbf{\$}\mathbf{60}$. Determine the supply function and inverse supply function for good $\mathbf{X}$. Graph the inverse supply function.

Short answer

1. The unit supplied by the product $\mathrm{x}$is$930$ units
2. The unit supplied by the product$\mathrm{x}$ is 0. It is not possible to supply the existing price with negative quantity supplied.
3. $\frac{{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}}{2}+135={\mathrm{P}}_{\mathrm{x}}$

Step-by-step solution

Given Information: 

1. ${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}600\mathrm{and}{\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}60$
   
2. ${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}80\mathrm{and}{\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}60$
   
3. ${\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}60$
   

Finding the unit supplied by the product in the first situation:

a.

To find the unit supplied by the product $\mathrm{x}$if ${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}600\mathrm{and}{\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}60$we need to use the formula –

${\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=-30+2{\mathrm{P}}_{\mathrm{x}}-4{\mathrm{P}}_{\mathrm{z}}$

Substituting the values in the equation, we get -

$\begin{array}{c}=(-30+2\times \mathrm{\$}600)-(4\times \mathrm{\$}60)\\ {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=(-30+2\times \mathrm{\$}600)-(4\times \mathrm{\$}60)\\ =-30+1200-240\\ =1200-270\\ =390\end{array}$

Thus, the unit supplied by the product X is 930 units.

Finding the unit supplied by the product in the second situation: 

b.

To find the unit supplied by the product X when ${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}80\mathrm{and}{\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}60$, we use the formula as -

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=-30+2{\mathrm{P}}_{\mathrm{x}}-4{\mathrm{P}}_{\mathrm{z}}\\ =-30+2\times 80-4\times 60\\ {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=-30+2\times 80-4\times 60\\ =-30+160-240\\ =-110\end{array}$

The unit supplied by the product x is zero due to negative value. It is not possible to supply the existing price with negative quantity supplied.

Step 4: Finding the inverse demand

c.

We find the unit supplied by the product$\mathrm{x}$ if ${\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}60$by using the formula as

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}=-30+2{\mathrm{P}}_{\mathrm{x}}-4{\mathrm{P}}_{\mathrm{z}}\\ =-30+2{\mathrm{P}}_{\mathrm{x}}-4\times 60\\ =-30+2{\mathrm{P}}_{\mathrm{x}}-240\\ =2{\mathrm{P}}_{\mathrm{x}}-270\\ 2{\mathrm{P}}_{\mathrm{x}}=270+{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}\\ {\mathrm{P}}_{\mathrm{x}}=\frac{{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}+270}{2}\\ {\mathrm{P}}_{\mathrm{x}}=\frac{{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}}{2}+135\end{array}$

Thus, the required inverse demand function is ${\mathrm{P}}_{\mathrm{x}}=\frac{{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{s}}}{2}+135$ .