Isoelastic function

In

For an elasticity coefficient ${\displaystyle r}$ (which can take on any real value), the function's general form is given by

${\displaystyle f(x)={kx^{r}},}$

where ${\displaystyle k}$ and ${\displaystyle r}$ are constants. The elasticity is by definition

${\displaystyle {\text{elasticity}}={\frac {df(x)}{dx}}{\frac {x}{f(x)}}={\frac {d{\text{ln}}f(x)}{d{\text{ln}}x}},}$

which for this function simply equals r.

Derivation

Elasticity of demand is indicated by

${\displaystyle {r}={\frac {dQ}{dP}}{\frac {P}{Q}}}$,

where r is the elasticity, Q is quantity, and P is price.

Rearranging gets us:

${\displaystyle {\frac {r}{P}}{dP}={\frac {1}{Q}}{dQ}}$

Then integrating

${\displaystyle \int {\frac {r}{P}}{dP}=\int {\frac {1}{Q}}{dQ}}$

${\displaystyle r\ln(P)+C=\ln(Q)}$

Simplify

${\displaystyle e^{\ln(P)r+C}=e^{ln(Q)}}$

${\displaystyle (e^{\ln(P)})^{r}e^{C}=Q}$

${\displaystyle kP^{r}=Q}$

${\displaystyle Q(P)=kP^{r}}$

Examples

Demand functions

An example in microeconomics is the constant elasticity demand function, in which p is the price of a product and D(p) is the resulting quantity demanded by consumers. For most goods the elasticity r (the responsiveness of quantity demanded to price) is negative, so it can be convenient to write the constant elasticity demand function with a negative sign on the exponent, in order for the coefficient ${\displaystyle r}$ to take on a positive value:

${\displaystyle D(p)={kp^{-r}},}$

where ${\displaystyle r>0}$ is now interpreted as the unsigned magnitude of the responsiveness.^[1] An analogous function exists for the

Utility functions in the presence of risk

The constant elasticity function is also used in the theory of choice under

• Constant elasticity of substitution
• Power function

References