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| In [[mathematics]], the '''elasticity''' or '''point elasticity''' of a positive [[differentiable function]] ''f'' of a positive variable (positive input, positive output)<ref>The elasticity can also be defined if the input and/or output is consistently negative, or simply away from any points where the input or output is zero, but in practice the elasticity is used for positive quantities.</ref> at point ''a'' is defined as<ref name=sydsaeter>{{cite book |authorlink=Knut Sydsæter |last=Sydsaeter |first=Knut |last2=Hammond |first2=Peter |title=Mathematics for Economic Analysis |location=Englewood Cliffs, NJ |publisher=Prentice Hall |year=1995 |pages=173–175 |isbn=013583600X }}</ref>
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| :<math>Ef(a) = \frac{a}{f(a)}f'(a)</math>
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| :<math>=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\frac{a}{f(a)}=\lim_{x\to a}\frac{f(x)-f(a)}{f(a)}\frac{a}{x-a}=\lim_{x\to a}\frac{1- \frac{f(x)}{f(a)}}{1-\frac{x}{a}}\approx \frac{\%\Delta f(a)}{\%\Delta a} </math>
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| or equivalently
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| :<math>Ef(x) = \frac{d \log f(x)}{d \log x}.</math>
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| It is thus the ratio of the relative (percentage) change in the function's output <math>f(x)</math> with respect to the relative change in its input <math>x</math>, for infinitesimal changes from a point <math>(a, f(a))</math>. Equivalently, it is the ratio of the infinitesimal change of the logarithm of a function with respect to the infinitesimal change of the logarithm of the argument.
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| The elasticity of a function is a constant <math>\alpha</math> if and only if the function has the form <math>f(x) = C x ^ \alpha</math> for a constant <math>C>0</math>.
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| The elasticity at a point is the limit of the [[arc elasticity]] between two points as the separation between those two points approaches zero.
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| The concept of elasticity is widely used in [[economics]]; see [[elasticity (economics)]] for details.
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| ==Rules==
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| Rules for finding the elasticity of products and quotients are simpler than those for derivatives. Let ''f, g'' be differentiable. Then<ref name=sydsaeter />
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| :<math>E ( f(x) \cdot g(x) ) = E f(x) + E g(x)</math>
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| :<math>E \frac{f(x)}{g(x)} = E f(x) - E g(x)</math> | |
| :<math>E ( f(x) + g(x) ) = \frac{f(x) \cdot E(f(x)) + g(x) \cdot E(g(x))}{f(x) + g(x)} </math> | |
| :<math>E ( f(x) - g(x) ) = \frac{f(x) \cdot E(f(x)) - g(x) \cdot E(g(x))}{f(x) - g(x)} </math>
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| The derivative can be expressed in terms of elasticity as
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| :<math>D f(x) = \frac{E f(x) \cdot f(x)}{x}</math> | |
| Let ''a'' and ''b'' be constants. Then
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| :<math>E ( a ) = 0 \ </math>
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| :<math> E ( a \cdot f(x) ) = E f(x) </math>,
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| :<math> E (b x^a) = a \ </math>.
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| ==Estimating point elasticities==
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| In economics, the [[elasticity of demand|price elasticity of demand]] refers to the elasticity of a [[demand function]] ''Q''(''P''), and can be expressed as (dQ/dP)/(Q(P)/P) or the ratio of the value of the [[marginal concepts|marginal function]] (dQ/dP) to the value of the average function (Q(P)/P). This relationship provides an easy way of determining whether a demand curve is elastic or inelastic at a particular point. First, suppose one follows the usual convention in mathematics of plotting the independent variable (P) horizontally and the dependent variable (Q) vertically. Then the slope of a line tangent to the curve at that point is the value of the marginal function at that point. The slope of a [[ray (geometry)|ray]] drawn from the origin through the point is the value of the average function. If the absolute value of the slope of the tangent is greater than the slope of the ray then the function is elastic at the point; if the slope of the secant is greater than the absolute value of the slope of the tangent then the curve is inelastic at the point.<ref>{{cite book |last=Chiang |last2=Wainwright |title=Fundamental Methods of Mathematical Economics |edition=4th |pages=192–193 |publisher=McGraw-Hill |year=2005 |location=Boston |isbn=0070109109 }}</ref> If the tangent line is extended to the horizontal axis the problem is simply a matter of comparing angles formed by the lines and the horizontal axis. If the marginal angle is greater than the average angle then the function is elastic at the point; if the marginal angle is less than the average angle then the function is inelastic at that point. If, however, one follows the convention adopted by economists and plots the independent variable ''P'' on the vertical axis and the dependent variable ''Q'' on the horizontal axis, then the opposite rules would apply.
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| The same graphical procedure can also be applied to a [[supply function]] or other functions.
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| ==Semi-elasticity==
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| A semi-elasticity (or semielasticity) gives the percentage change in ''f(x)'' in terms of a change (not percentage-wise) of ''x''. Algebraically, the semi-elasticity S of a function ''f'' at point ''x'' is <ref>{{cite book |title=Introductory Econometrics: A Modern Approach |edition=2nd |authorlink=Jeffrey Wooldridge |first=Jeffrey |last=Wooldridge |publisher=South-Western |ISBN=0-324-11364-1 |year=2003 |page=656}}</ref><ref>{{cite book|title=The theory of monetary institutions |first=Lawrence Henry |last=White |ISBN=0-631-21214-0 |year=1999 |location=Malden |publisher=Blackwell |page=148}}</ref>
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| :<math>Sf(x) = \frac{1}{f(x)}f'(x) = \frac{d \ln f(x)}{d x}</math>
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| An example of semi-elasticity is [[modified duration]] in bond trading.
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| The term "semi-elasticity" is also sometimes used for the change if ''f(x)'' in terms of a percentage change in ''x''<ref>http://www.stata.com/help.cgi?margins</ref> which would be
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| :<math>\frac{d f(x)}{d\ln(x)}=\frac{d f(x)}{dx}x</math>
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| ==See also==
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| * [[Arc elasticity]]
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| * [[Elasticity (economics)]]
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| * [[Homogeneous function]]
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| ==References==
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| {{Reflist}}
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| *{{cite journal |first=Yves |last=Nievergelt |title=The Concept of Elasticity in Economics |journal=SIAM Review |volume=25 |issue=2 |year=1983 |pages=261–265 |doi=10.1137/1025049 }}
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| {{DEFAULTSORT:Elasticity Of A Function}}
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| [[Category:Functions and mappings]]
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| [[Category:Mathematical economics]]
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Greetings! I am Myrtle Shroyer. Doing ceramics is what her family members and her enjoy. My family members lives in Minnesota and my family members loves it. Managing people has been his working day job for a whilst.
Feel free to surf to my blog :: http://inspirationpedi.com/