Question

The demand function for good $\mathbf{X}$is ln${\mathbf{InQ}}_{\mathbf{x}}^{\mathbf{d}}\mathbf{=}\mathbf{a}\mathbf{+}{\mathbf{bInP}}_{\mathbf{x}}\mathbf{+}\mathbf{cInM}\mathbf{+}\mathbf{e}$, where ${\mathbf{P}}_{\mathbf{X}}$is the price of good X and M is income. Least squares regression reveals that$\widehat{\mathbf{a}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{42}\mathbf{,}\widehat{\mathbf{b}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{18}\mathbf{;}\mathbf{}\mathbf{and}\mathbf{}\widehat{\mathbf{c}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{34}$

a. If $\mathbf{M}\mathbf{=}\mathbf{55}\mathbf{,}\mathbf{000}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{39}$, compute the own price elasticity of demand based on these estimates. Determine whether demand is elastic or inelastic.

b. If $\mathbf{M}\mathbf{=}\mathbf{55}\mathbf{,}\mathbf{000}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{39}$, compute the income elasticity of demand based on these estimates. Determine whether $\mathbf{X}$is a normal or inferior good

Short answer

1. The own price elasticity of demand for Good$\mathrm{X}$ is -2.18, $\left|\mathrm{E}\right|⟩1$, hence elastic.
2. The income elasticity of demand for Good$\mathrm{X}$ is 0.34,$\left|\mathrm{E}\right|⟨1$ hence, inelastic and the good is inferior.

Step-by-step solution

Step 1: Given Information:

${\mathrm{InQ}}_{\mathrm{x}}^{\mathrm{d}}=\mathrm{a}+{\mathrm{bInP}}_{\mathrm{x}}+\mathrm{cInM}+{\mathrm{e}}_{\mathrm{x}}$,$\mathrm{M}=55,000;{\mathrm{P}}_{\mathrm{x}}=4.39;\mathrm{and}\widehat{\mathrm{c}}=0.34$

Step 2: To find the own price elasticity:

a.

Demand function for Good$\mathrm{X}$is log- linear that the coefficient of a variable gives that elasticity of demand with respect to its variable.

Own price elasticity is equated as

role="math" localid="1658494814365" ${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}={\mathrm{\beta }}_{\mathrm{x}},\mathrm{so},\mathrm{if}\mathrm{M}=55,000;{\mathrm{P}}_{\mathrm{x}}=4.39;\mathrm{and}\widehat{\mathrm{b}}=-2.18$

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}={\mathrm{\beta }}_{\mathrm{x}}=-2.18$

Thus, the own price elasticity of demand for Good$\mathrm{X}$ is${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=-2.18$

Hence, the own price elasticity of demand for Good $\mathrm{X}$ is elastic because, $\left|\mathrm{E}\right|⟩1$.

Step 3: To find the income price elasticity of demand:

b.

Demand function for Good$\mathrm{X}$is log- linear that the coefficient of a variable gives that elasticity of demand with respect to its variable.

Income price elasticity is equated as

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{M}}={\mathrm{\beta }}_{\mathrm{M}},\mathrm{so},\mathrm{if}\mathrm{M}=55,000;{\mathrm{P}}_{\mathrm{x}}=4.39;\mathrm{and}\widehat{\mathrm{c}}=0.34,$

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}={\mathrm{\beta }}_{\mathrm{M}}=0.34$

Thus, the Income elasticity of demand for Good $\mathrm{X}$is${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{M}}=0.34$

Hence, the Income elasticity of demand for Good$\mathrm{X}$ is inelastic because $\left|\mathrm{E}\right|⟨1$.

Since a change in income will not change the consumption of the good much, hence, it is an inferior good.