Isoelastic function
Encyclopedia
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.
is the constant elasticity demand function
, in which x is the price of a product and f(x) 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.
, 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
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 
→ ∞).
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...
, an isoelastic function, sometimes constant elasticity function, is a function that exhibits a constant elasticity
Elasticity (economics)
In economics, elasticity is the measurement of how changing one economic variable affects others. For example:* "If I lower the price of my product, how much more will I sell?"* "If I raise the price, how much less will I sell?"...
, i.e. has a constant elasticity coefficient
Elasticity Coefficient
Elasticity Coefficients are used in Physics, Economics, Chemistry, or more generally in mathematics as a definition of point elasticity: the article below applies to Chemical/Biochemical Elasticity Coefficients....
. The elasticity is the ratio of the percentage change in the dependent variable to the percentage causative change in the independent variable
Independent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
, 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 bywhere
and are constants. The elasticity is by definitionwhich for this function simply equals r.
Demand functions
An example in microeconomicsMicroeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...
is the constant elasticity demand function
Demand curve
In economics, the demand curve is the graph depicting the relationship between the price of a certain commodity, and the amount of it that consumers are willing and able to purchase at that given price. It is a graphic representation of a demand schedule...
, in which x is the price of a product and f(x) 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.Utility functions in the presence of risk
The constant elasticity function is also used in the theory of choice under risk aversionRisk aversion
Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....
, which usually assumes that risk-averse decision-makers maximize the expected value of a concave
Concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...
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
Stock market
A stock market or equity market is a public entity for the trading of company stock and derivatives at an agreed price; these are securities listed on a stock exchange as well as those only traded privately.The size of the world stock market was estimated at about $36.6 trillion...
in a portfolio
Modern portfolio theory
Modern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
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 → ∞).