The Multiplicative Deivative as a Measue o Elasticity i Ecoomics - Itoductio Feado Códova-Lepe Istituto de Ciecias Básicas Uivesidad Católica del Maule codova@ucmcl I [] it was itoduced a kid o deivative o multiplicative type This deivative pemits to compae elative chages betwee a depedet positive vaiable ad the idepedet espective vaiable that it is also positive Thee we deied a ew opeatio o the positive eal umbes that makes commutative the aise to a powe opeatio Give a ad b i l(b) ] itoducig a b a as a secod opeatio this is ate the usual multiplicatio we do o the positive eal umbes a odeed ad complete ield ie a model o the eal umbes ield that lives i its iteio Fo a uctio :] ] we deie the multiplicative deivative o i ] [ as the limit i it eists Q ( ( h) l( h) h ) lim ( ) () Some o the picipal popeties that chaacteize the calculus iduced with this ew deivative ae listed i the ollowig theoem Theoem : Let g :] ] be uctios such that the deivatives Q ( ) ad Qg ( ) eist o some ] [ the: ( i) I is a costat uctio the Q ( ) ( ii) I ( ) with a eal umbe the Q ( ) e (iii) We have β β Q ( g )( ) ( Q ( )) ( Qg( )) o all ad β eal umbes (iv) We have Q( g ) ( Qg) ( Q ) ( g ) So i a > the Q ( a ) a ( v) We have Q( So i a > the + g) ( Q ) Q( + a) ( Q ) + g + a ( Qg) g + g
Othe impotat esult that pemits to calculate the multiplicative deivative o compouded uctios is the aalogous o the Chai Rule; it is the theoem that ollows Theoem : Let g :] ] be uctios such that the deivatives Q g( )) ad Qg ( ) eist o some ] [ the: l(qg ) ( Q ( g) ) Qg ( Q ( g)) Q ( o g) ( This deivative geeates a calculus that we called Popotioal Calculus PC Thee is a isomophism betwee PC ad the Dieetial Calculus DC This idetiicatio puts the PC ito de Newto s Calculus It is hoped the that the ew deivative that coespod to a multiplicative poit o view o the vaiatio's epesetatios (i cotast to the additive oe which is the usual pespective) will help i the solutio o poblems o the taditioal deivative Sice what it was said we ca ie that i the PC is a image that keeps the stuctue o the DC the ew calculus also is a whole that cotais aothe calculus i its iteio geeatig i this mae a iiite chai o deivatives ad its espective calculus We cite the ollowig esult that is vey useul techically because it ties both deivatives Theoem 3: Let :] ] ad some positive eal umbes The Q ) eists i oly i D ( ) eists i that case ( ) D ( ) l( Q ( )) () The multiplicative deivative has i some cases techical advatages o calculus picipally i the teatmet o uctios costucted by mi o powes multiplicatios ad thei ivese opeatios I this wok we peted to show some possible ad eplicit applicatios o the PC i Ecoomics Moe pecisely i sectio we itoduce a elatio that appeas betwee the multiplicative deivative ad the kow cocept o the elasticity The elasticity cocept is too much eteded ad useul a shot glossay cosides: Coss-pice elasticity o demad Coss-pice elasticity o supply Elastic (elasticity > ) Icome elasticity o demad Ielastic (elasticity < ) Ieio goods (icome elasticity < ) Luuies (icome elasticity > ) Necessity (icome elasticity < ) Nomal goods (icome elasticity > ) Peectly elastic cuves Peectly ielastic cuves Pice elasticity o demad Pice elasticity o supply Pice elasticity ad Uit elastic (elasticity ) We thik that the cocept o elasticity deseves to have i popety a associate calculus ad the PC is a good cadidate I sectio 3 we coect the ew deivative with the homogeeous uctios essetially we pesets a ew epesetatio o the Eule Theoem Moeove we do some special emaks o two impotat classes o these uctios we epeset the Cobb Douglas Fuctios as liea tasomatios ad o Costat Elasticity o Substitutio Fuctios (o to calculate the substitutio elasticity) we get a omula that is easy to emembe (
- The Elasticity Elasticity ca be thought as the cocept that measues the esposiveess o oe vaiable i espose to aothe vaiable I ecoomics the best measue o this esposiveess is the popotioal o the pecet chage i oe vaiable elative to the popotioal chage i aothe vaiable hee we ee to positive vaiables I :] ] is a uctio that elates two vaiables ad y by y () so that > ad y > ae give such that y ( ) the the popotioal chages i idepedet ad depedet vaiables ae / ad y / y espectively whee ad y y y The compaisos betwee these popotioal chages depedet vesus idepedet ae usually got by the quotiet opeatio y / y (3) / Fiig > this umbe is a uctio o so that it detemies a aveage elasticity o o the iteval + ] i > o + ] i < Fo to obtai a [ [ istataeous elasticity (poit elasticity) o at it is ecessay to do Epessio (3) i this case covets to: dy '( ) ( ) (4) y d ( ) Let be positive i has bee calculated the the lieaizatio o at this is y / '( ) i allows us to estimate the pecet chage y / y give a pecet chage / with the omula y (5) y Fo eample suppose tha i a couty is the uctio that elates the quatity demaded o meat ( y ) with the icome ( ) i i some > the icome elasticity o demad is 8 ad the pecetage chage i icome is 5% What ca we hope i the pecetage chage i demad? The aswe is y 8 5% 9% y We ca coect epessio (4) with () to wite the istataeous elasticity o at this is l( Q ( )) (6) So the elasticity is the atual logaithm o the multiplicative deivative o at A diect applicatio o (6) ad the ive items o Theoem is that they pemit to get quickly
the ollowig elatios amog the elasticity o a uctio ad the elasticity o its compoets Theoem 4: Let g :] ] be uctios such that ad g eist o some ] the: ( i) I is a costat uctio the l( Q ( )) l() ( ii) I ( ) with a eal umbe the l( Q ( )) l( e ) (iii) We have β β l[( Q ( )) ( Qg( )) ] g + β g o all ad β eal umbes (iv) We have l[( Qg) ( Q ) ( g )] ( ) [ g + g( )] g So i a > the l( Q( a )) l( a ) l( ) ( v) We have So i a > the a a g + g + + g + g g l[( Q ) ( Qg) ] + + g g g + a + a l[( Q ) ] + a The elatio o the chai ule o to calculate elasticity has the om l( Qg ) l[( Q ( g)) ] o g g ( ) g All these omulas the Theoem 4 ad the chai ule ae well kow; hee we do ot wat to show a uoldig oigiality oly to emak the acility i mae o to do the calculatios the immediateess o to obtai them kowig well the calculus ules o the PC which is the cost that we must pay I the PC i y () whee :] ] the the elative chage i the idepedet vaiable om to ca be measued by similaly o the idepedet vaiable this is y with y y These magitudes ca be compaed usig the ivese o the secod ( ) opeatio o the ield (] ) so we get y y l( / )
Epessio that i the limit whe is deoted by Q ( ) Clealy it also measue the esposiveess o vaiable y i espose to the vaiable ; so we pemit us to call it PCelasticity Let y () be the demad uctio o meat at a icome i i some > the icome PC-elasticity o demad is 65 ad the icome gowth is 5% the o to estimate the elative chage o the demad we use y y Q ( ) 65 l(5) 5 l(65) 9 So we coclude that the demad icease i 9% Theeoe ow the elasticity is a kid o deivative we do ot eed to lea a omula to calculate it A classic eecise is to id what uctios :] ] have costat elasticity With the additive deivative this poblem passes o solvig the odiay dieetial equatio '( ) + ( ) (7) With ou deivative the espective poblem is to solve Q ( ) e (8) It is vey easy to get the solutio o (7) by the method o sepaated vaiables but o solvig (8) it is eough to thik i uctios with multiplicative costat deivative seeig the ist thee popeties i Theoem we have ( ) > 3- Homogeeous Fuctios A uctio elatio: :] ] is a homogeeous uctio o ode (degee) i it satisies the ( t) t ( ) o all t > (9) It is easy to pove tha the ( ) k moeove this is equivalet to Q ( ) Q( k) Q( ) e Whe we wok with uctios o seveal vaiables this is :] ] the the coespodig deiitio is the same epessio (9) but with a vecto i ] The Eule s Homogeeous Fuctio Theoem that elates the degee o homogeeity with the patial deivatives aims that: ' ( ) + ' ( ) + L + ' ( ) i ' i ( ) () I tems o the multiplicative deivative the elatio () takes the om: Q ) Q ( ) Q ( ) e ( K ()
Whee () with i K is the patial multiplicative deivative o at this is Q i / l( h) [ ( K ih K ) / ( )] as h We ca ote some simple advatages o () cooted with () o istace: the autoomy o this epessio with espect to the idepedet vaiable the possibility o vebalizig as a homogeeous uctio has costat the poduct o all its patial elasticities This setece is ot diicult to memoize Thee ae two classical amilies with high utility i ecoomic theoy o homogeeous uctios they ae: a) The Cobb-Douglas Fuctios ad b) The CES (Costat Elasticity o Substitutio) Fuctios A uctio :] ] is a Cobb Douglas Fuctio (CD-uctio) i ( L L ) k whee k > o all i L The elatio () o a CD-uctio takes the om i e K e e so it is immediate that + L+ The way o to thik ad chaacteize a cocept i its elatio with othes has diect coectio with the cotet whee it is thought Fo istace we ca aim that a CDuctio is a liea tasomatio betwee vecto spaces deied o the ield (] ) ( R + ) It is hee the behid easo o to speak sometimes o the logaithmic- liea model Fo that we deie ove the set ] the poduct ( L ) ( y L y) ( y L y) It is tivial to pove that we have a commutative goup ad i this goup the eutal elemet take the om ( L) ad the ivese o a elemet ( L ) is ( L ) It is also possible to deie the scala poduct o a elemet L ) by a positive eal umbe ] by the equality ( ( L ) ( l( ) L l( ) ) This ew scala poduct satisies: ( β ) ( β ) ( y) ( ) ( y) ( β ) ( ) ( β ) ad that e o all β > ad y ] All these popeties pemit us to claim (] ) is a vecto space ove the itoduced ield o the positive eal umbes With the sets ] [ ad ] [ thought as vecto spaces o (] [ ) each liea tasomatio T :] ] is such that T[( ) ( β y)] ( T ( )) ( β T ( y)) this is the image o a liea combiatio o vectos is the liea combiatio o the images o such vectos Sice [ ( e L)] [ ( e L)] L[ ( L e)] to deie T i a uique mae it is suiciet to deie the images o the base omed by ( e L) ( e L) ( L e) i thei images ae espectively L the we have T ) ( ) ( ) L( ) (
β β β ie T ( ) L whee β i l( i) i K So that a CD-uctio ca be see i this cotet as a liea tasomatio Pehaps could be iteestig the study i the ecoomic ield o the mati epesetatio o a liea tasomatio betwee spaces o the om ] picipally i elatio with the tascedetal logaithm uctioal 5- CES Fuctios β β Fo a CD-uctio o two vaiables ( β > > it is deied o ) i i ( y) ( ) ] : ( ) y each > the cotou o at this value as the set C y { } The gaph o this cotou is a cocave level cuve whe we put the vaiable i uctio o the vaiable Fo to obtai the slopes we calculates implicit additive deivatives i the elatio that deie the cotou as a uctio o gettig the equality I Ecoomics the people is iteestig i to kow how this slope vaies alog o the cuve o that it is studied its itesectio with the staight lie C ad it is studied the vaiatio o such slope whe C vaies Fo a bette measue o the sesibility to the chages ad obviatig the sig it is cosideed the elasticity this is ' ' ' ( ) ' The elasticity o substitutio is deied asσ A eample o a uctio with costat elasticity o substitutio is k ( ) () v ρ δ δ + ρ ρ It ca be poved that this uctio has costat elasticity o substitutio σ (3) + ρ I i the give deiitio o substitutio elasticity we epess the Newtoias deivatives i tems o the aalogous multiplicative deivatives we have: σ + l Q l( Q l( Q ( / ) ) )
Note the similaity o this last epessio which is satisied o ay homogeeous uctio with the epessio (3) that was obtaied o a paticula case o CES uctio the uctio deied i () It is also possible to combie both deivatives to measue some dieet kids o esposiveess 6- Reeeces Códova-Lepe F Fom quotiet opeatio towad a popotioal calculus Joual o Mathematics Game Theoy ad Algeba (4) Accepted Madde P Cocavidad y optimizació e micoecoomía Aliaza Editoial S A Madid (987)