Isoelastic function

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 $r$ (which can take on any real value), the function's general form is given by

f(x)={kx^{r}},

where $k$ and $r$ are constants. The elasticity is by definition

{\text{elasticity}}={\frac {df(x)}{dx}}{\frac {x}{f(x)}}={\frac {d{\text{ln}}f(x)}{d{\text{ln}}x}},

which for this function simply equals r.

Derivation

Elasticity of demand is indicated by

${r}={\frac {dQ}{dP}}{\frac {P}{Q}}$ ,

where r is the elasticity, Q is quantity, and P is price.

Rearranging gets us:

${\frac {r}{P}}{dP}={\frac {1}{Q}}{dQ}$

Then integrating

$\int {\frac {r}{P}}{dP}=\int {\frac {1}{Q}}{dQ}$

$r\ln(P)+C=\ln(Q)$

Simplify

$e^{\ln(P)r+C}=e^{ln(Q)}$

$(e^{\ln(P)})^{r}e^{C}=Q$

$kP^{r}=Q$

$Q(P)=kP^{r}$

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 $r$ to take on a positive value:

D(p)={kp^{-r}},

where $r>0$ is now interpreted as the unsigned magnitude of the responsiveness.^[1] 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, 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)={\frac {1}{1-\gamma }}x^{1-\gamma }

where x is wealth and $1-\gamma$ is the elasticity, with $\gamma >0$ , $\gamma$ ≠ 1 referred to as the constant coefficient of relative risk aversion (with risk aversion approaching infinity as $\gamma$ → ∞).

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

Read other articles:

Bellan Roos

Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada November 2022. Bellan RoosRoos pada 1927Lahir(1901-05-25)25 Mei 1901Nynäshamn, SwediaMeninggal8 April 1990(1990-04-08) (umur 88)Stockholm, SwediaPekerjaanPemeranTahun aktif1933-1979 Bellan Roos (25 Mei 1901 – 8 April 1990) adalah seorang p...

دوري السوبر الألباني 1945

يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (نوفمبر 2019) دوري السوبر الألباني 1945 تفاصيل الموسم دوري السوبر الألباني النسخة 8 البلد ألبانيا التاريخ بداي...

Negara bagian di Nigeria

Peta negara bagian di Nigeria Negara bagian adalah wilayah administratif tingkat satu di Nigeria. Lihat pula ISO 3166-2:NG Templat:Negara bagian di Nigeria Artikel bertopik Nigeria ini adalah sebuah rintisan. Anda dapat membantu Wikipedia dengan mengembangkannya.lbs

Nanfan Treaty

1701 land cession by the Iroquois Confederacy to the British Crown Nanfan TreatyDeed from the Five Nations to the King, of their Beaver Hunting GroundMap showing the approximate range of the Beaver Hunting Grounds described in the Nanfan Treaty.SignedJuly 19, 1701 (1701-07-19)LocationAlbany, New YorkAmendmentSeptember 14, 1726 (1726-09-14)Parties Iroquois Confederacy John Nanfan Full text Nanfan Treaty at Wikisource Deed from the Five Nations to the King, o...

Carl Switzer

Carl Switzer è Alfa Alfa in Simpatiche canaglie (dal 1935 al 1940) Carlton Dean Switzer, noto anche con lo pseudonimo di Alfalfa (Paris, 7 agosto 1927 – Mission Hills, 21 gennaio 1959), è stato un attore statunitense. È conosciuto per le sue apparizioni come attore bambino e personaggio principale, nel ruolo di Alfalfa (o Alfa Alfa), nel noto serial cinematografico degli anni trenta Our Gang (Simpatiche canaglie). Proseguì la carriera di attore anche da ragazzo, per poi diventare un all...

Pupuk Indonesia Niaga

PT Pupuk Indonesia NiagaSebelumnyaPT Mesin dan Gaya Electro (1964 - 1977)PT Mesin Gaya Electro dan Trading (1977 - 2023)JenisPerseroan terbatasIndustriPerdagangan dan konstruksiDidirikan1940; 84 tahun lalu (1940)KantorpusatJakarta, IndonesiaWilayah operasiIndonesiaTokohkunciHilman Taufik[1](Direktur Utama)Eko Cahyanto[1](Komisaris Utama)ProdukCatJasaPerdagangan pupuk, bahan bakar, pelumas, karung, dan pewarnaKonstruksi ketenagalistrikanPendapatanRp 1,72 triliun (2021)[...

Горностай

У этого термина существуют и другие значения, см. Горностай (значения). Горностай Научная классификация Домен:ЭукариотыЦарство:ЖивотныеПодцарство:ЭуметазоиБез ранга:Двусторонне-симметричныеБез ранга:ВторичноротыеТип:ХордовыеПодтип:ПозвоночныеИнфратип:Челюстнороты...

Parti russe (Grèce)

Pour les articles homonymes, voir Parti russe. Parti russeHistoireFondation 1827Dissolution 1865CadreType Early Greek party, parti politiqueSiège AthènesPays GrèceOrganisationIdéologie Étatisme, populisme, monarchisme, russophilie, conservatisme socialPositionnement Droitemodifier - modifier le code - modifier Wikidata Le Parti russe (en grec moderne : Ρωσικό Κóμμα), ou parti napiste (Ναπαίοι) était un des trois premiers partis politiques créés en Grèce ...

Namgyal Tsemo Monastery

Namgyal Tsemo MonasteryNamgyal Tsemo MonasteryReligionAffiliationTibetan BuddhismSectGelugLocationLocationLeh district, Ladakh, IndiaCountryIndiaLocation within IndiaGeographic coordinates34°10′3″N 77°35′24″E / 34.16750°N 77.59000°E / 34.16750; 77.59000ArchitectureFounderTashi Namgyal Namgyal Tsemo Monastery or Namgyal Tsemo Gompa is a Buddhist monastery in Leh city of Leh district, Ladakh, northern India.[1] Founded by King Tashi Namgyal (1555-157...

1963 National Challenge Cup

Football tournament season 1963 National Challenge CupDewar Challenge CupTournament detailsCountry USADefending championsNew York HungariaFinal positionsChampionsUkrainian Nationals (3rd title)Runner-upLos Angeles Armenians1964 CONCACAF Champions' CupUkrainian Nationals← 19621964 → The 1963 National Challenge Cup was the 50th edition of the USSFA's annual open soccer championship. The Philadelphia Ukrainians defeated the Los Angeles Armenian to win. Final June 2, 1...

Theater in Ukraine

This article needs to be updated. The reason given is: the category title is inappropriate; should be UKRAINIAN THEATRE. Please help update this article to reflect recent events or newly available information. (September 2013) Ukrainian Discourse Theatre in Lviv, Ukraine. 1864. Part of a series onUkrainians List of Ukrainians Culture Architecture Art Cinema Cuisine Dance Literature Music Sport Theater Languages and dialects Ukrainian Russian Crimean Tatar Balachka Surzhyk Religion Eastern Ort...

Scythische talen

Scythisch Gesproken in Sarmatië, Scythië, Sistan, Scythia Minor, Alanië Uitgestorven in de oudheid Taalfamilie Indo-Europees Indo-Iraans Oost-Iraans Scytisch Varianten Westelijk: Alaans Oostelijk: Hotanees Taalcodes ISO 639-1 geen ISO 639-2 geen ISO 639-3 xsc - Scythisch xln - Alaans oos - Oud-Ossetisch Portaal Taal De Scythische talen zijn een groep van Oost-Iraanse talen uit de klassieke en de late oudheid, gesproken in een uitgestrekt gebied van Eurazië genaamd ...

Football Club Arzignano Valchiampo

FC Arzignano ValchiampoCalcio Grifoni, Giallocelesti, Oro-celesti Segni distintiviUniformi di gara Casa Trasferta Colori sociali Giallo, celeste SimboliGrifone Dati societariCittàArzignano Nazione Italia ConfederazioneUEFA Federazione FIGC CampionatoSerie C Fondazione1920 Rifondazione2011Presidente Lino Chilese Allenatore Alessandro Bruno StadioTommaso Dal Molin(1 690 posti) Sito webwww.arzignanovalchiampo.it Palmarès Stagione in corso Si invita a seguire il modello di voce Il Foo...

Teramo Calcio 1976-1977

Voce principale: Società Sportiva Teramo Calcio. Teramo CalcioStagione 1976-1977Sport calcio Squadra Teramo Allenatore Maurizio Bruno Presidente Carmine Pedicone Serie C11º posto nel girone B. Maggiori presenzeCampionato: Esposito, Izzo (37) Miglior marcatoreCampionato: Pulitelli (15) 1975-1976 1977-1978 Si invita a seguire il modello di voce Questa pagina raccoglie le informazioni riguardanti il Teramo Calcio nelle competizioni ufficiali della stagione 1976-1977. Rosa N. Ruolo Calcia...

Georgics

Poem by Virgil Georgic redirects here. For other uses, see Georgic (disambiguation). Georgics Book III, shepherd with flocks, Roman Virgil.The Georgics (/ˈdʒɔːrdʒɪks/ JOR-jiks; Latin: Georgica [ɡeˈoːrɡɪka]) is a poem by Latin poet Virgil, likely published in 29 BCE.[1] As the name suggests (from the Greek word γεωργικά, geōrgiká, i.e. agricultural (things))[2] the subject of the poem is agriculture; but far from being an example of peaceful rural p...

Bosco (drag queen)

American drag performer BoscoBosco at RuPaul's DragCon LA, 2022Born (1993-06-11) June 11, 1993 (age 31)Great Falls, Montana, U.S.Other namesBlair ConstantinoTelevisionRuPaul's Drag Race (season 14)Websitehereisbosco.com Blair Constantino[1] (born June 11, 1993), known by the stage name Bosco,[2] is an American drag performer who competed on season 14 of RuPaul's Drag Race in 2022. Early life Bosco was raised by musicians in Great Falls, Montana.[2] She[a&...

Armoise

Pour les articles homonymes, voir Artemisia. Artemisia Artemisia Artemisia schmidtiana (en), l'Armoise naine.Classification de Cronquist (1981) Règne Plantae Sous-règne Tracheobionta Division Magnoliophyta Classe Magnoliopsida Sous-classe Asteridae Ordre Asterales Famille Asteraceae GenreArtemisiaL., 1753 Classification APG III (2009) Classification APG III (2009) Clade Angiospermes Clade Dicotylédones vraies Clade Noyau des Dicotylédones vraies Clade Astéridées Clade Campanulidé...

Hesychasm

Eastern Orthodox contemplative prayer Part of a series onPalamismGregory Palamas History Hesychast controversy Fifth Council of Constantinople Synod of Jerusalem Neo-Palamism Theology Essence–energies distinction Hesychasm Theoria Theotokos Theosis Tabor Light Hypostatic union Apophatic theology Influences and precursors Plato Paul the Apostle Plotinus Neoplatonism Basil of Caesarea Gregory of Nazianzus Gregory of Nyssa John Chrysostom Pseudo-Dionysius John of Damascus Photios Symeon the Ne...

San Carlo alle Quattro Fontane

Church in Rome, ItalySan Carlo alle Quattro FontaneChurch of Saint Charles at the Four FountainsSan Carlo alle Quattro FontaneFaçade of Church of San Carlo alle Quattro Fontane by Francesco BorrominiClick on the map to see marker.41°54′07″N 12°29′27″E / 41.90181°N 12.49074°E / 41.90181; 12.49074LocationVia del Quirinale 23 00184, RomeCountryItalyLanguage(s)ItalianDenominationCatholicTraditionRoman RiteReligious orderTrinitariansWebsitewww.sancarlino-borrom...

Copisa

No debe confundirse con Compañía Perú International. Sede de Copisa en Hospitalet de Llobregat Copisa es una empresa constructora fundada el 1959 con sede en Hospitalet de Llobregat (Barcelona) que opera en el sector de las infraestructuras, en el ámbito de la obra civil, edificación y rehabilitación. Historia 1959-1970 Los orígenes de la empresa se encuentran en el proceso histórico de electriﬁcación de Cataluña, liderado en la década de los 50 por FECSA. En 1959 nació Construc...

PROFILPELAJAR.COM