Question

Suppose the demand function for a firm’s product is given by

$\mathit{l}\mathit{n}\mathbf{}{\mathit{Q}}_{\mathbf{x}}^{\mathbf{d}}\mathbf{=}\mathbf{7}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{}\mathit{l}\mathit{n}\mathbf{}{\mathit{P}}_{\mathbf{x}}\mathbf{+}\mathbf{2}\mathbf{}\mathit{l}\mathit{n}\mathbf{}{\mathit{P}}_{\mathbf{y}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{}\mathit{l}\mathit{n}\mathbf{}\mathit{M}\mathbf{+}\mathit{l}\mathit{n}\mathbf{}\mathit{A}$

where ${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash (}\mathbf{15}\mathbf{;}{\mathbf{P}}_{\mathbf{y}}\mathbf{=}\mathbf{\backslash )}\mathbf{6}\mathbf{;}\mathbf{M}\mathbf{=}\mathbf{40}\mathbf{,}\mathbf{000}\mathbf{}\mathbf{and}\mathbf{}\mathbf{A}\mathbf{=}\mathbf{350}$.

a. Determine the own price elasticity of demand, and state whether demand is elastic, inelastic, or unitary elastic.

b. Determine the cross-price elasticity of demand between good X and good Y, and state whether these two goods are substitutes or complements.

c. Determine the income elasticity of demand, and state whether good X is a normal or inferior good.

d. Determine the own advertising elasticity of demand

Short answer

1. $\left|{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}\right|>1$.The own price elasticity of demand is greater than 1. It means the demand is elastic.
2. $\left|{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{y}}}\right|>0$. Both the goods are substitutes.
3. $\left|{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{M}}}\right|>0$. Good is inferior.
4. ${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{A}}=1$

Step-by-step solution

Given information 

It is given that${\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}15;{\mathrm{P}}_{\mathrm{y}}=\mathrm{\$}6;\mathrm{M}=40,000\mathrm{and}\mathrm{A}=350$ , and demand firm product is ${\mathrm{InQ}}_{\mathrm{x}}^{\mathrm{d}}=7-1.5{\mathrm{InP}}_{\mathrm{x}}+2{\mathrm{InP}}_{\mathrm{y}}-0.5\mathrm{InM}+\mathrm{InA}$.

Step 2: To find the own price elasticity demand function 

b.

Use the following equation to determine the own price elasticity demand:

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

Therefore, the coefficient for${\mathrm{InP}}_{\mathrm{x}}$${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}={\mathrm{\beta }}_{\mathrm{x}}=-1.5$ .

Thus, $\left|{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}\right|>1$. The own price elasticity of demand is greater than $1$. The demand is elastic.

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

b.

We use the following equation to determine the cross-price elasticity demand:

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{y}}}={\mathrm{\beta }}_{\mathrm{y}}$

We determine the coefficient of variables using equation ${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{y}}}={\mathrm{\beta }}_{\mathrm{y}}=2$.

Thus,$\left|{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{y}}}\right|>0$ as the price of Good $\mathrm{X}$and $\mathrm{Y}$Good are increasing. Hence, both are substitute goods.

Step 4: To find the income-price elasticity of demand

c.

To determine the income elasticity demand, we use equation ${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{M}}}={\mathrm{\beta }}_{\mathrm{M}}$.

It is given that$\mathrm{X}$ is an inferior good to determine the sign coefficient of variables as

$\mathrm{InM}$.

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{M}}}=-0.5$

Therefore,$\left|{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{M}}}\right|>0$ . Good$\mathrm{X}$ is inferior.

Step 5: To find the own advertising elasticity of demand:

d.

To determine the own advertising elasticity demand, we use equation${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{A}}={\mathrm{\beta }}_{\mathrm{A}}$ .

Therefore,${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{A}}=1$ is the own advertising demand.