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
(which can take on any real value), the function's general form is given by
:
where
and
are constants. The elasticity is by definition
:
which for this function simply equals ''r''.
Derivation
Elasticity of demand is indicated by
,
where r is the elasticity, Q is quantity, and P is price.
Rearranging gets us:
Then integrating
Simplify
Examples
Demand functions
An example in
microeconomics is the constant elasticity
demand function
In economics, a demand curve is a graph depicting the relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be used either for ...
, 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
to take on a positive value:
:
where
is now interpreted as the unsigned magnitude of the responsiveness.
An analogous function exists for the
supply curve.
Utility functions in the presence of risk
The constant elasticity function is also used in the theory of choice under
risk aversion
In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more c ...
, which usually assumes that risk-averse decision-makers maximize the expected value of a
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity of a ...
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
Portfolio may refer to:
Objects
* Portfolio (briefcase), a type of briefcase
Collections
* Portfolio (finance), a collection of assets held by an institution or a private individual
* Artist's portfolio, a sample of an artist's work or a c ...
are independent of the scale of the decision-maker's wealth. The constant elasticity utility function in this context is generally written as
:
where ''x'' is wealth and
is the elasticity, with
,
≠ 1 referred to as the constant coefficient of relative risk aversion (with risk aversion approaching infinity as
→ ∞).
See also
*
Constant elasticity of substitution
Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions. Several economists have featured in the topic and have contributed in the final finding of the constant. They include Tom McK ...
*
Power function
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
References
External links
Constant Elasticity Demand and Supply Curves
{{DEFAULTSORT:Isoelastic Function
Mathematical economics