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Modelyuvannya Economy - Vіtlіnsky VV

8. ROZDІL MODELІ POVEDІNKI VIROBNIKІV

Maksimіzatsіya pributku - Basic kriterіy on yaky orієntuyutsya virobniki. Ale tse not єdiny kriterіy. Maksimіzatsіya stream pributku guilty spіvvіdnositisya Zi strategіchnim forecast rozvitku fіrmi toscho.

8.1. Model fіrmi

Nekhay virobnicha FIRMA vipuskaє one product (chi bagato produktіv, ale z postіynoyu structure). Rіchny Key infrastructure from natural-rechovіy formі X - tse Quantity odinits product of one mind (chi Quantity bagatonomenklaturnih agregatіv).

Vikoristanі resources: L - alive Praca (from viglyadі serednoї chiselnostі zaynyatih for Year chi vіdpratsovanih for Person of Year-Godin) K - zasobi pratsі (osnovnі virobnichі Fund); M - Objects pratsі (vitrachene for Year palivo, energіya, sirovina, materіali, komplektuvalnі virobi toscho).

Leather s agregovanih vidіv resursіv (Prace, foundations, materіali) Got Pevnyi Quantity rіznovidіv.

Poznachimo vector stovpchik mozhlivih obsyagіv vitrat rіznih vidіv resursіv by x = (x 1, ..., xn) ?. Todі tehnologіya fіrmi viznachatimetsya її virobnichoyu funktsієyu, yak virazhaє phone reception mіzh vitratami resursіv i Key infrastructure:

X = F (x). (8.1)

Pripuskaєtsya gіpoteza scho F (x) dvіchі neperervno diferentsіyovana neoklasichna i before Well matrix її other pohіdnih Je vіd'єmno viznachenoyu.

Yakscho w = (w 1, ..., wj, ..., wn ) - vector-row tsіn resursіv and p - Rate produktsії, the dermal vector x vitrat vіdpovіdaє Prybutok:

P (x) = pF (x) - wx . (8.2)

In (8.2) R = pX = pF (x) - vartіst rіchnogo Key infrastructure fіrmi abo її rіchny dohіd, C = wx - vitrati virobnitstva chi vartіst vitrat resursіv for the Year.

Yakscho not vvoditi іnshih obmezhen, okrіm nevіd'єmnih vitrat resursіv, the problem for a maximum pributku viglyadu dial:

(8.3)

Tse task nelіnіynogo programuvannya n s minds nevіd'єmnostі x? 0 neobhіdnimi minds її rozv'yazannya Je minds Kuhn-Tucker:

(8.4)

Yakscho in optimal rozv'yazku vikoristovuyutsya All types resursіv, tobto x *> 0, the minds (8.4) matimut viglyad:

(8.5)

abo

tobto in optimalnіy tochtsі vartіst boundary Product danogo resource guilty dorіvnyuvati yogo tsіnі.

Taqiy samy (for forms) rozv'yazok Got the maximization Key infrastructure Quest obsyagu vitrat

(8.6)

Tse task nelіnіynogo programuvannya s one lіnіynim obmezhennyam i minds nevіd'єmnostі zmіnnih.

Pobuduymo funktsіyu Lagrange:

L (x, l) = F (x) + l (C - wx),

Teper maksimіzuєmo її for nevіd'єmnostі zmіnnih minds.

For tsogo neobhіdno, dwellers vikonuvalis minds Kuna Thackeray:

(8.7)

Yak bachimo, minds (8.7) tsіlkom zbіgayutsya s (8.4), Yakscho poklast

Butt Key infrastructure odnoproduktovoї fіrmi zadaєtsya virobnichoyu funktsієyu Cobb-Douglas:

Viznachimo Maximum Key infrastructure, Yakscho on Orenda fondіv i pay pratsі vidіleno 150 penny. od., vartіst Orenda odinitsі fondіv wk = 5 penny. od., bid Salary w L = 10 penny. od. / folk.

Yakoyu bude a boundary rule zamіni one zaynyatogo funds in optimalnіy tochtsі?

Rozv'yazannya. Oskіlki F (0, L) = L (K, 0) = 0, then the optimal rozv'yazku K *> 0, L *> 0, that minds (8.7) gain the viglyadu:

(8.8)

abo in Nashomu vipadku:

Podіlivshi Pershe rіvnyannya on the other, maєmo:

.

Pіdstavivshi Tsey viraz in the minds:

znaydemo

Rozv'yazannya mozhna proіlyustruvati geometrically. Fig. 8.1 zobrazhenі іzokosti (lіnії postіynih vitrat for C = 50, 100, 150) that іzokvanti (lіnії postіynih vipuskіv for X = 25.2; 37.8).

Іzokosti toil takі rіvnyannya:

5 10 K + L = C = const.

Іzokvanti -

Fig. 8.1. Іzokosti postіynih vitrat that іzokvanti postіynih vipuskіv

In optimalnіy tochtsі K * = 20, L * = 5 іzokvanta X * = 37,8 іzokosta that C = 150, passing through scho qiu point dotikayutsya, more zgіdno s (8.8) normalі to Tsikh curve zadanі gradієntami kolіnearnі.

Norma zamіschennya pratsі funds in optimalnіy tochtsі:

tobto one pratsyuyuchy Mauger Buti zamіneny EYAD odinitsyami fondіv.

Rozv'yazuyuchi modelі fіrmi problem (8.3) for a maximum pributku, znahodimo єdiny optimally nabіr resursіv x *> 0 (rozglyadaєtsya vipadok, if OAO All incoming resources to set). Tsomu set vіdpovіdaє єdine vitrat values: C * = wx *.

Rozv'yazuєmo modelі fіrmi problem (8.6) for a maximum pributku Quest vitrat C *. Yakscho F (x) - neoklasichna, the optimal rozv'yazku And Tsey rozv'yazok єdiny.

So rank, s one side,

MAZDA іnshogo -

Oskіlki

the

ale , to

Through those scho rozv'yazok zadachі (8.3) єdiny, the

Otzhe, Yakscho task at maximum Prybutok Got єdiny rozv'yazok x *> 0, the їy vіdpovіdaє task at maximum Key infrastructure Quest vitrat C * = wx *, And Stay Got Taqiy samy rozv'yazok, yak i Persha (div. Fig. 8.1):

Geometric Location tochok dotiku іzokost that іzokvant for rіznih values vitrat C viznachaє dovgotermіnovy Way rozvitku fіrmi the X (C), tobto pokazuє, yak zrostatime (spadatime) Key infrastructure, Yakscho vitrati zrostut (zmenshatsya). Oskіlki tsya zalezhnіst monotone, then іsnuє Auburn monotone funktsіya vitrat C = C (X).

Oskіlki X (C) - Maximum Key infrastructure Quest vitrat C, vitrati C (X), SSMSC vіdpovіdayut tsomu maximum Key infrastructure X - mіnіmalnі and optimally obsyag Key infrastructure znovu Well viznachaєtsya minds for maximum pributku:

. (8.9)

Prirіvnyuyuchi zero pohіdnu

bachimo, scho in optimalnіy tochtsі granichnі vitrati dorіvnyuyut tsіnі Key infrastructure:

Okrіm of a maximum of pributku dosyagaєtsya

(bo ).

Rozglyanmo spіvvіdnoshen n (8.5):

Tsі spіvvіdnoshennya shaping can Buti rozv'yazanі vіdnosno h okolі optimalnoї point x *, Yakscho yakobіan de

Oznachaє Tse, scho guilty Buti vіdmіnnim od zero gessіan virobnichoї funktsії (Ale H vіd'єmno viznacheny, to dіysno ) Todі

(8.10)

abo

Tsі n rіvnyan zadayut funktsіyu popitu (for resources), vіdshukanі for Relief modelі povedіnki fіrmi. Funktsії popitu on resources can takozh Know experiental for Relief metodіv statistics matematichnoї for vіdpovіdnimi vibіrkovimi danimi.

Funktsіya offers -

.

Podіbno to rіvnyannya Slutskogo scho pokazuє reaktsіyu spozhivacha on zmіni tsіn tovarіv, analogіchnі rіvnyannya opisuyut reaktsіyu virobnika on zmіnu tsіn Key infrastructure i resursіv.

Over the minds of tasks tsіn p, w povedіnka virobnika viznachaєtsya such spіvvіdnoshennyam (usogo (n + 1) spіvvіdnoshen):

(8.11)

Nekhay teper Rate Key infrastructure zmіnilasya chi zmіnilasya Cena resursіv, abo i those i those.

1. Reaktsіya virobnika on zmіnu tsіni Key infrastructure.

Diferentsіyuєmo (8.11) for p:

abo poznachennі in the matrix:

de - Vector-row, - Vector stovpchik, abo

(8.12)

Rіvnyannya (8.12) yavlyaє him reaktsіyu virobnika (zmіnu Key infrastructure) (zmіnu popitu to resources) on zmіnu tsіni Key infrastructure p.

2. Reaktsіya virobnika on zmіnu tsіn resursіv.

Nekhay zmіnilasya Cena k-th resource wk, todі diferentsіyuєmo rіvnyannya (8.11) per wk:

(8.13)

Yakscho poznachiti

then n (n + 1) rіvnyannya (8.13) in matrichnіy formі be written in such a rank (tse reaktsіya virobnika on zmіnu tsіn resursіv):

(8.14)

3. Reaktsіya virobnika on odnochasnu zmіnu tsіni Key infrastructure that tsіni resursіv. Poєdnannya (8.12) that (8.14) daє basic matrix rіvnyannya teorії fіrmi:

(8.15)

yak pokazuє reaktsіyu virobnika on odnochasnu zmіnu tsіni Key infrastructure i tsіn resursіv.

Rozv'yazuyuchi (8.15) vіdnosno zmіni Key infrastructure i zmіni popitu on resources , Otrimuєmo:

(8.16)

Skoristavshis rule obernennya block matrix, maєmo:

Pіdstavlyayuchi ostannіy viraz from (8.16), otrimaєmo system rіvnyan fіrmi vіdnosno for Change Key infrastructure i popitu on resources:

. (8.17)

Pershe rіvnyannya the system (8.17) pokazuє, yak zmіnitsya Key infrastructure for zrostannya tsіni on produktsіyu fіrmi. Oskіlki Hessian H vіd'єmno viznachena, the H -1 - takozh, to

otzhe,

(8.18)

tobto Zi zrostannyam tsіni Key infrastructure obsyag Key infrastructure produktsії zrostaє.

So rank, .

(8.19)

Ale (For neoklasichnoї funktsії granichnі produkti dodatnі) to obov'yazkovo deyakі tobto zrostannya tsіni Key infrastructure given to zrostannya popitu on deyakі resources.

Resource l th mean nazivayut malotsіnnim, Yakscho h (8.17) can be seen (a friend that third grupa rіvnyan), scho abo in rozgornutomu viglyadі -

(8.20)

Besides zrostannya tsіni on produktsіyu zumovlyuvatime pіdvischennya (znizhennya) popitu on okremі Vidi vitrat, Yakscho pіdvischennya tsіni on Tsey view resursіv lead to skorochennya (zrostannya) obsyagіv optimum Key infrastructure. Zokrema, zbіlshennya tsіni on malotsіnny resource spriyatime zbіlshennyu Key infrastructure.

Pіdstavlyayuchi (8.20) from (8.19), otrimaєmo

s ago viplivaє scho tobto zrostannya tsіni on deyaky resursіv view zooms skorochennya Key infrastructure.

Zgіdno s (8.17)

Besides the matrix - Vіd'єmno viznachena, otzhe, tobto pіdvischennya tsіni on deyaky resource zavzhdi generally produces to decline popitu to Demba, otzhe, krivі popitu Je spadnimi.

Oskіlki matrix - Symmetrically, the

(8.21)

tobto vpliv zmіni tsіni the l - th share on zmіnu popitu to the j-th resource i zmіna tsіni by j-th resource on zmіnu popitu the l th resource odnakovі.

Vitrati j th i l th vidіv resursіv Je vzaєmozamіnyuvanimi (vzaєmodopovnyuyuchimi) Yakscho toscho.