Question

Show how to determine elasticities from linear and log-linear demand functions.

Short answer

Elasticity is measured through the change in responsiveness of one variable to another.

Step-by-step solution

Definition of Elasticity

It depicts the measurement of responsiveness of one variable to the percentage change in other variables.

Linear Demand Function

It represents the demand function which is the linear combination of price income and other variables.

It is written in the form,

${\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}={\text{α}}_{\text{0}}+{\text{α}}_{\text{x}}{\mathrm{P}}_{\mathrm{x}}+{\text{α}}_{\text{y}}{\mathrm{P}}_{\mathrm{y}}+{\text{α}}_{\text{M}}\mathrm{M}+{\text{α}}_{\text{H}}\mathrm{H}$

Where Coefficients of price represent whether the commodities are substitutes or complements.

M = Income.

Determining elasticities from linear demand functions

Own price Elasticity:${\text{E}}_{{\text{Q}}_{\text{x}}{\text{,P}}_{\text{x}}}={\text{α}}_{\text{x}}\frac{{\mathrm{P}}_{\mathrm{x}}}{{\mathrm{Q}}_{\mathrm{x}}}$

Cross price Elasticity: ${\text{E}}_{{\text{Q}}_{\text{x}}{\text{,P}}_{\mathrm{y}}}={\text{α}}_{\mathrm{y}}\frac{{\mathrm{P}}_{\mathrm{y}}}{{\mathrm{Q}}_{\mathrm{x}}}$’

Income Elasticity:${\text{E}}_{{\text{Q}}_{\text{x}}\text{,M}}={\text{α}}_{\mathrm{M}}\frac{\mathrm{M}}{{\mathrm{Q}}_{\mathrm{x}}}$

Log-Linear Demand Function

It represents the demand function which is the linear function of logarithms of prices, incomes, etc.

It is written in the form,

${\mathrm{lnQ}}_{\mathrm{x}}^{\mathrm{d}}={\mathrm{\beta }}_{\text{0}}+{\mathrm{\beta }}_{\text{x}}{\mathrm{lnP}}_{\mathrm{x}}+{\mathrm{\beta }}_{\text{y}}{\mathrm{lnP}}_{\mathrm{y}}+{\mathrm{\beta }}_{\text{M}}\mathrm{lnM}+{\mathrm{\beta }}_{\text{H}}\mathrm{lnH}$

Where,${\mathrm{\beta }}_0=\ln \left(\mathrm{c}\right)$

${\mathrm{\beta }}_{\mathrm{i}}$= arbitrary real numbers and it determines whether the commodities are substitutes or complements.

Determining elasticities from log-linear demand function

If the demand function for a certain commodity X and is log linear then, the elasticities can be written as,

Own price Elasticity:${\text{E}}_{{\text{Q}}_{\text{x}}{\text{,P}}_{\text{x}}}={\mathrm{\beta }}_{\mathrm{x}}$

Cross price Elasticity:${\text{E}}_{{\text{Q}}_{\text{x}}{\text{,P}}_{\mathrm{y}}}={\mathrm{\beta }}_{\mathrm{y}}$

Income Elasticity:${\text{E}}_{{\text{Q}}_{\text{x}}\text{,M}}={\mathrm{\beta }}_{\mathrm{M}}$

These are calculated by taking the anti-logarithm of the log-linear demand function.