Question

The demand for good $\mathbf{X}$is given by ${\mathbf{Q}}_{\mathbf{x}}^{\mathbf{d}}\mathbf{=}\mathbf{6}\mathbf{,}\mathbf{000}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{P}}_{\mathbf{x}}\mathbf{-}{\mathbf{P}}_{\mathbf{y}}\mathbf{+}\mathbf{9}{\mathbf{P}}_{\mathbf{z}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{10}}\mathbf{M}$Research shows that the prices of related goods are given by ${\mathbf{P}}_{\mathbf{y}}\mathbf{=}\mathbf{\backslash (}\mathbf{6}\mathbf{,}\mathbf{500}\mathbf{,}{\mathbf{P}}_{\mathbf{z}}\mathbf{=}\mathbf{\backslash )}\mathbf{100}$, while the average income of individuals consuming this product is $\mathbf{M}\mathbf{=}\mathbf{\backslash (}\mathbf{70}\mathbf{,}\mathbf{000}$.

1. Indicate whether goods$\mathbf{Y}$and$\mathbf{Z}$are substitutes or complements for good$\mathbf{X}$.
2. Is$\mathbf{X}$an inferior or a normal good?
3. How many units of good$\mathbf{X}$will be purchased when${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash )}\mathbf{5}\mathbf{,}\mathbf{230}$?
4. Determine the demand function and inverse demand function for good$\mathbf{X}$. Graph the demand curve for good$\mathbf{X}$

Short answer

1. .Goods Y and Z are complementary.
2. X is a normal good because the earnings coefficient is positive. A rise in profits, for example, will lead to an increase in consumption
3. ${\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=4785$
   
4. ${\mathrm{P}}_{\mathrm{x}}=14,800-2{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}$

Step-by-step solution

Given Information:

${\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=6,000-\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}-{\mathrm{P}}_{\mathrm{y}}+9{\mathrm{P}}_{\mathrm{z}}+\frac{1}{10}\mathrm{M}$

role="math" localid="1658488962131" ${\mathrm{P}}_{\mathrm{y}}=\mathrm{\$}6,500{\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}100$

$\mathrm{M}=\mathrm{\$}70,000$

Finding the nature of good

a.

When the cost of good rises, so does the quantity of demand for the good. The good coefficient also becomes negative. As the good coefficient is positive, an increase in good's price will result in an increase in the amount desired of purchased goods. So, goods Y and Z are complementary.

Finding normality or inferiority

b.

X is a normal good because the earnings coefficient is positive. A rise in profits, for example, will lead to an increase in consumption.

Finding quantity

c.

${\mathrm{P}}_{\mathrm{y}}=\mathrm{\$}6,500{\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}100$

$\mathrm{M}=\mathrm{\$}70,000$

The quantity of Good$\mathrm{X}$ is

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=6,000-\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}-{\mathrm{P}}_{\mathrm{y}}+9{\mathrm{P}}_{\mathrm{z}}+\frac{1}{10}\mathrm{M}\\ {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=6,000-\frac{1}{2}(5,230)-6,500+9\left(100\right)+\frac{1}{10}(70,000)\\ {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=6,000-2,615-6,500+900+7,000\\ {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=4,785\end{array}$

Thus, required quantity is 4785 units.

Finding find the demand function and inverse demand function using graph

d.

${\mathrm{P}}_{\mathrm{y}}=\mathrm{\$}6500$, ${\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}100$ ,$\mathrm{M}=\mathrm{\$}70,000$

The demand function for $\mathrm{X}$is

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=6,000-\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}-{\mathrm{P}}_{\mathrm{y}}+9{\mathrm{P}}_{\mathrm{z}}+\frac{1}{10}\mathrm{M}\\ {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=6,000-\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}-6,500+9\left(100\right)+\frac{1}{10}(70,000)\\ =6,000-\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}-6,500+900+7,000\\ =7,400-\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}\end{array}$

The demand function is ${\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=7,400-\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}$

To find the inverse demand function to solve ${\text{P}}_{\mathrm{x}}$using the demand function

$\begin{array}{c}\frac{1}{2}{\mathrm{P}}_{\mathrm{x}}=7,400-{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}\\ 2\times \frac{1}{2}{\mathrm{P}}_{\mathrm{x}}=2(7,400-{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}})\\ {\mathrm{P}}_{\mathrm{x}}=2\times 7,400-2\times {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}\\ =14,800-2{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}\end{array}$

Therefore, the inverse demand function is ${\mathrm{P}}_{\mathrm{x}}=14,800-2{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}$

The demand curve graph as shown as