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Modelyuvannya Economy - Vіtlіnsky VV

7.2. Rіvnyannya Slutskogo

Rozglyanmo, yak zmіnitsya popit spozhivacha scho viznachaєtsya Modell (7.3):

Yakscho zmіnitsya Cena s tovarіv one.

Nekhay Cena n th product zrosla on apn, todі zgіdno s (7.6) on popit Leather goods zmіnitsya as follows:

Ale, oskіlki (7.6) - Tse deprivation Inshyj records (7.4) i (7.5), then for rіvnyannya obsessively, diferentsіyuyuchi for pn tsі spіvvіdnoshennya, tobto:

(7.7)

System z (n + 1) lіnіynih rіvnyan (7.7) vіdnosno (n + 1) nevіdomih in matrichnіy formі is written in such a rank (bar oznachaє transponuvannya, p - vector tsіn row, U - Hessian matrix, x * - vector stovpchik popitu on the product):

(7.8)

Matrix, wrap up matritsі rіvnyan (7.8), Got viglyad:

de m = - (pU -1 p? ) -1> 0.

Tom rozv'yazok Sistemi rіvnyan (7.8) in matrichnіy formі Got viglyad:

Іndeks matritsі n y (in the bow) oznachaє scho taken її n th stovpchik.

Otzhe, zbіlshennya (zmіna) tsіni of the n-th product led to takoї zmіni popitu for the goods:

. (7.9)

Teper dotsіlno z'yasuvati zmіst warehouses, scho incoming rіvnyannya in (7.9).

Zmіna popitu for zbіlshennya tsіni s kompensatsієyu

Rozglyanmo TAKE zbіlshennya income on dM, yak b kompensuvalo spozhivachevі zbіlshennya tsіni on dpn (at the n-th product benefit).

Zgіdno s theorem spozhivannya oznachaє Tse, scho korisnіst spozhivacha zbereglasya on poperednomu rіvnі, tobto:

Maєmo vikoristati rіvnyannya (7.5) h yakogo otrimaєmo:

Zvіdsi maєmo scho:

abo, residually, Umov postіynostі korisnostі:

(7.10)

Now we can viznachiti dM, vikoristovuyuchi (7.4) that (7.10):

tobto dohіd zrіs rіvno nastіlki, skіlki neobhіdno Bulo b dodatkovo vitratiti spozhivachevі pridbannya on the n-th product in poperednomu obsyazі for zrostannya tsіni the value of dpn.

Diferentsіyuvannya rіvnyannya (7.5) of the pn lead to quiet themselves rezultatіv, scho i ranіshe. Tom for kompensovanoї zmіni tsіni otrimaєmo takі rіvnyannya for :

viplivaє h

(7.11)

SSMSC in matrichnіy formі pluck viglyadu:

(7.12)

Rozv'yazok Sistemi rіvnyan (7.12) znovu Well znahodimo for Relief obernenoї matritsі:

Otzhe, zbіlshennya tsіni s kompensatsієyu income lead to takoї zmіni popitu:

(7.13)

Tobto other warehouses in pravіy chastinі (7.9) - tse zmіna popitu, Yakscho zrostannya tsіni n th product on dpn kompensuєtsya zbіlshennyam income on

Rozglyanmo zmіnu popitu for zmіni income.

Nekhay dohіd zmіnyuєtsya on dM, todі popit zmіnitsya as follows:

Rіvnyannya for znovu Well after otrimaєmo, diferentsіyuyuchi for spіvvіdnoshennya M (7.4), (7.5):

(7.14)

abo in matrichnіy formі:

(7.15)

that

.

zvіdki maєmo:

(7.16)

Ob'єdnuyuchi (7.9) (7.13) (7.16) otrimaєmo TAKE rіvnyannya Slutskogo, yak Yea nitty gritty teorії korisnostі:

(7.17)

A oskіlki vivchaєtsya zmіna popitu for zrostannya tsіni of the n-th product, scho not kompensuєtsya pіdvischennyam income, the other warehouses in (7.17) (h vіd'єmnim sign) yakraz i znіmaє boxed prirіst popitu scho viklikany kompensuyuchim zrostannyam income.

To dwellers skoristatisya rіvnyannyam Slutskogo, vivchimo vlastivostі matritsі:

Tsya matrix symmetrically, symmetrically oskіlki Je matrix U -1, and Stop symmetrically vnaslіdok simetrichnostі matritsі U.

Matrix H vіd'єmno napіvviznachena scho oznachaє: zHz? ? 0 for whether yakogo vector-row z.

Rozglyanemo vipadok, if z = a p, a? 0 todі

de ).

Nekhay teper napryamok z is not zbіgaєtsya napryamkom s p (z a p tobto of whether yakogo a?), Todі z You can have viglyadі taxes:

z = a p + v, v = z - a p,

de (CE number), n? 0 (Yakscho b n = 0, z = a p).

W this filed z maєmo:

and that

oskіlki U -1 vіd'єmno viznachena.

Yakscho take as the z vector of z = (0, ..., 1, 0, ..., 0), tobto vector-row, usі Elements yakogo, okrіm i- th, dorіvnyuyut zero, and the i- th yelement dorіvnyuє 1, then

zHz? = Hii <0,

OAO All tobto dіagonalnі Elements matritsі H Je vіd'єmnimi. that

(7.18)

Otzhe, navіt for kompensovanogo zrostannya tsіni product popit on Tsey goods mustache still spadaє.

Goods i nazivayut tsіnnim, Yakscho Zi zrostannyam income popit on Demba zrostaє , Malotsіnnim - Yakscho .

Oskіlki zgіdno s (7.14)

(7.19)

the tsіnnі Flea obov'yazkovo іsnuyut.

Popit on tsіnny goods spadaє for zrostannya tsіni to Demba, tse bezposeredno viplivaє s rіvnyannya Slutskogo for (i th) product:

Zgіdno s (7.11)

Besides obov'yazkovo znaydetsya Taqiy goods l, for yakogo

Tobto zmenshennya popitu on the i-th item lead to zrostannya popitu the l th commodity.

Takі Flea nazivayut vzaєmozamіnyuvanimi, napriklad, tvarinnі Jiri i Roslyn olіya.

Yakscho Well , The product is the i m utvoryuyut vzaєmodopovnyuvalnu couple (kompensovane zrostannya tsіni gasoline sprichinyuє decline popitu yak on gasoline, so i on avtomobіlі).

Product l nazivayut zamіnnikom gross product i, Yakscho

Funktsіya popitu Got vlastivіst valovoї zamіnnostі, Yakscho Zi zrostannyam tsіni on whether yaky product i popit to Rasht produktіv not znizhuєtsya:

Yakscho Well Then funktsіya popitu Got vlastivіst strong gross zamіschennya. You can bring, scho funktsіya popitu, porodzhena funktsієyu korisnostі

Got vlastivostі strong gross zamіschennya.

Efekta zamіschennya that income for pіdvischennya (znizhennya) tsіni one іz tovarіv іlyustruyut Fig. 7.1 is the 7.2 vіdpovіdno.

Fig. 7.1. Efekt zamіschennya that efekt income for pіdvischennya tsіni

Fig. 7.2.Efekt zamіschennya that efekt income for znizhennya tsіni

Efekt income polyagaє in zmіnі spozhivannya vnaslіdok zmіni real income, yak vinikla through zmіnu tsіn.

Efekt zamіschennya polyagaє in zmіnі spozhivannya vnaslіdok zmіni vіdnosnih tsіn.