In mathematical economics, an isoelastic function, sometimes constant elasticity function, is a function that exhibits a constant elasticity, i.e. has a constant elasticity coefficient. The elasticity is the ratio of the percentage change in the dependent variable to the percentage causative change in the independent variable, in the limit as the changes approach zero in magnitude.
For an elasticity coefficient
r
{\displaystyle r}
where
k
{\displaystyle k}
which for this function simply equals r.
Derivation
Elasticity of demand is indicated by
r
=
d
Q
d
P
P
Q
{\displaystyle {r}={\frac {dQ}{dP}}{\frac {P}{Q}}}
where r is the elasticity, Q is quantity, and P is price.
Rearranging gets us:
r
P
d
P
=
1
Q
d
Q
{\displaystyle {\frac {r}{P}}{dP}={\frac {1}{Q}}{dQ}}
Then integrating
∫
r
P
d
P
=
∫
1
Q
d
Q
{\displaystyle \int {\frac {r}{P}}{dP}=\int {\frac {1}{Q}}{dQ}}
r
ln
(
P
)
+
C
=
ln
(
Q
)
{\displaystyle r\ln(P)+C=\ln(Q)}
Simplify
e
ln
(
P
)
r
+
C
=
e
l
n
(
Q
)
{\displaystyle e^{\ln(P)r+C}=e^{ln(Q)}}
(
e
ln
(
P
)
)
r
e
C
=
Q
{\displaystyle (e^{\ln(P)})^{r}e^{C}=Q}
C
P
r
=
Q
{\displaystyle CP^{r}=Q}
Q
(
p
)
=
k
P
r
{\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 r {\displaystyle r} to take on a positive value:
D ( p ) = k p − r , {\displaystyle D(p)={kp^{-r}},}where
r
>
0
{\displaystyle r>0}
Utility functions in the presence of risk
The constant elasticity function is also used in the theory of choice under risk aversion, which usually assumes that risk-averse decision-makers maximize the expected value of a concave von Neumann-Morgenstern utility function. In this context, with a constant elasticity of utility with respect to, say, wealth, optimal decisions on such things as shares of stocks in a portfolio are independent of the scale of the decision-maker's wealth. The constant elasticity utility function in this context is generally written as
U ( x ) = 1 1 − γ x 1 − γ {\displaystyle U(x)={\frac {1}{1-\gamma }}x^{1-\gamma }}where x is wealth and
1
−
γ
{\displaystyle 1-\gamma }
See also
- Constant elasticity of substitution
- Power function
References
- ^ Simon, Carl P.; Blume, Lawrence (1994). Mathematics for Economists. New York: Norton. p. 67. ISBN 0393957330.
External links
- Constant Elasticity Demand and Supply Curves
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