Question

The demand curve for a product is given by${\mathit{Q}}_{\mathbf{x}}^{\mathbf{d}}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{200}\mathbf{-}\mathbf{3}\mathit{P}\mathit{x}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{1}\mathit{P}\mathit{z}$

where ${\mathbf{P}}_{\mathbf{z}}\mathbf{=}\mathbf{\backslash (}\mathbf{300}$.

a. What is the own price elasticity of demand when ${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash )}\mathbf{140}$? Is demand elastic or inelastic at this price? What would happen to the firm’s revenue if it decided to charge a price below $\mathbf{\backslash (}\mathbf{140}$?

b. What is the own price elasticity of demand when ${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash )}\mathbf{240}$? Is demand elastic or inelastic at this price? What would happen to the firm’s revenue if it decided to charge a price above ${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash (}\mathbf{240}$?

c. What is the cross-price elasticity of demand between good X and good Z when ${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash )}\mathbf{140}$? Are goods X and Z substitutes or complements?

Short answer

1. The own price inelasticity demand at${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}140$ is -0.56. The demand will be inelastic because the value of elasticity is$\left|{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}\right|<1$ , decrease in the price level below $\mathrm{\$}140$will decrease the total amount of revenue.
2. Based on the calculation the demand will be elastic the own price inelasticity of demand at${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}240$ is -1.6. Since,$\left|{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}\right|>1$ rise in the price level is$\mathrm{\$}240$ and decrease in the total amount of revenue.
3. The cross-price elasticity of demand is${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{z}}}=-0.04$ .Hence, Good X and Z are compliments.

Step-by-step solution

Step 1: To find the own price elasticity demand function at P=$140:

a.

Given the price and elasticity demand as${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}140,{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=1,200-3{\mathrm{P}}_{\mathrm{x}}-0.1{\mathrm{P}}_{\mathrm{z}}$

To find the current demand function as

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}[{\mathrm{P}}_{\mathrm{x}}=140,{\mathrm{P}}_{\mathrm{z}}=300]=1,200-3\times 140-0.1\times 300\\ =1,200-420-30\\ =750\end{array}$

The current demand as 750 units.

To find the own price elasticity demand function as

$\begin{array}{c}{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\frac{\partial {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}}{\partial {\mathrm{P}}_{\mathrm{x}}}\times \frac{{\mathrm{P}}_{\mathrm{x}}}{{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}}\\ {\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\frac{\partial (1,200-3{\mathrm{P}}_{\mathrm{x}}-0.1{\mathrm{P}}_{\mathrm{z}})}{\partial {\mathrm{P}}_{\mathrm{x}}}\times \frac{140}{750}\\ {\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=-3\times \frac{140}{750}\\ {\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=-0.56\end{array}$

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=-0.56$

Based on the calculation the demand will be inelastic the own price inelasticity demand at ${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}140$is -0.56. $\left|{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}\right|<1$.

Decreasing the price level below $140 will decrease the revenue. Since the demand is inelastic, decreasing the price will not change the demand much, thus decreasing its revenue.

Step 2: To calculate the price elasticity of demand at P=$240:

b.

Given the price and elasticity demand as

${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}240,{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=1,200-3{\mathrm{P}}_{\mathrm{x}}-0.1{\mathrm{P}}_{\mathrm{z}},{\mathrm{P}}_{\mathrm{z}}=\mathrm{\$}300.\mathrm{\$}$

To find the current demand function as

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}[{\mathrm{P}}_{\mathrm{x}}=240,{\mathrm{P}}_{\mathrm{z}}=300]=1,200-3\times 240-0.1\times 300\\ =1,200-720-30\\ =450\end{array}$

The current demand is$450$.

To calculate the price elasticity of demand as

$\begin{array}{c}{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\frac{\partial {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}}{\partial {\mathrm{P}}_{\mathrm{x}}}\times \frac{{\mathrm{P}}_{\mathrm{x}}}{{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}}\\ =\frac{\partial (1,200-3{\mathrm{P}}_{\mathrm{x}}-0.1{\mathrm{P}}_{\mathrm{z}})}{\partial {\mathrm{P}}_{\mathrm{x}}}\times \frac{240}{450}\\ =-3\times \frac{240}{450}\\ =-1.6\end{array}$

Based on the calculation the demand will be elastic, because the value of elasticity is greater than one,rise in the price level is$\$240$ and decrease in the total amount of revenue as consumers will reduce their purchase more rigorously to the change in price.

Step 3: To find the cross-price elasticity of demand: 

c.

Given that the demand for the cross-price elasticity demand is${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{y}}}={\mathrm{\beta }}_{\mathrm{y}}$

To find the cross-price elasticity of demand is

$\begin{array}{c}{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}[{\mathrm{P}}_{\mathrm{x}}=140,{\mathrm{P}}_{\mathrm{z}}=300]=1,200-3\times 140-0.1\times 300\\ =1,200-420-30\\ =750\end{array}$

The current demand as$750$units

$\begin{array}{l}\begin{array}{c}{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\frac{\partial {\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}}{\partial {\mathrm{P}}_{\mathrm{z}}}\times \frac{{\mathrm{P}}_{\mathrm{z}}}{{\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}}\\ =\frac{\partial (1,200-3{\mathrm{P}}_{\mathrm{x}}-0.1{\mathrm{P}}_{\mathrm{z}})}{\partial {\mathrm{P}}_{\mathrm{x}}}\times \frac{300}{750}\\ =-0.1\times \frac{300}{750}\\ =-0.04\end{array}\\ {\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{z}}}=-0.04\end{array}$

Thus, the cross-price elasticity demand is ${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{z}}}=-0.04$.

Goods X and Z compliments because the cross-price elasticity demand for Good X and Z is negative.