You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Ô(t) =7⋅ 10 6 +3⋅ 10 6 e –0,1t .<br />
ρ<br />
6<br />
Î÷åâèäíî, ùî limΦ () t = I = 7⋅10<br />
. Öå îçíà÷àº, ùî äëÿ<br />
t →∞ µ<br />
ðîçãëÿíóòîãî âèïàäêó ³ç çðîñòàííÿì ÷àñó âåëè÷èíà Ô çíà-<br />
÷íî çìåíøóºòüñÿ. Íå äóæå âò³øíà êàðòèíà. Àëå ç³ ñòðóêòóðè<br />
ð³âíÿííÿ ìîæíà çðîáèòè òàêèé âèñíîâîê: äëÿ òîãî ùîá<br />
ôîíä ï³äïðèºìñòâà íå çìåíøóâàâñÿ, òðåáà çä³éñíþâàòè ñòðàòåã³þ<br />
òàê, ùîá ³íâåñòèö³ÿ áóëà çì³ííîþ ³ çàäîâîëüíÿëà<br />
óìîâó<br />
1<br />
≥ Φ .<br />
7<br />
() () t<br />
I t<br />
11.9.3. Ìîäåëü Ñîëîó<br />
Ñòàí åêîíîì³êè, ÿê ºäèíå ö³ëå, çàäàºòüñÿ ï’ÿòüìà çì³ííèìè<br />
ñòàíó: Q — ê³íöåâèé ïðîäóêò, L — òðóäîâ³ ðåñóðñè,<br />
K — âèðîáíè÷³ ôîíäè, ² — ³íâåñòèö³¿, Ñ — ðîçì³ð íåâèðîáíè÷îãî<br />
ñïîæèâàííÿ.  ë³òåðàòóð³ ç åêîíîì³êè, â ðàìêàõ<br />
ìîäåë³ Ñîëîó, ï’ÿòü çãàäàíèõ âèùå çì³ííèõ ïîâ’ÿçàí³ ì³æ<br />
ñîáîþ òàêèìè ñï³ââ³äíîøåííÿìè:<br />
( ) ( ) 0<br />
C = 1 −ρ Q, 0 ≤ρ< 1; Q = Q K, L ; L = L e νt ;<br />
dK<br />
Q K,<br />
K( o) K0<br />
dt =ρ −µ = . (11.9.12)<br />
Òóò âåëè÷èíà ρ íàçèâàºòüñÿ íîðìîþ íàêîïè÷åííÿ ³ õàðàêòåðèçóº<br />
äîëþ ê³íöåâîãî ïðîäóêòó, ÿêó âèêîðèñòîâóþòü<br />
íà ³íâåñòèö³¿; âåëè÷èíà µ º êîåô³ö³ºíò âèáóòòÿ ôîíä³â. Áóäåìî<br />
ïðè öüîìó ââàæàòè, ùî âèðîáíè÷à ôóíêö³ÿ Q = Q(K, L)<br />
º îäíîð³äíîþ ôóíêö³ºþ ïåðøîãî ïîðÿäêó, òîáòî<br />
Q(λK, λL) =λQ(K, L) (ïðèðîäíà óìîâà äëÿ âèðîáíè÷èõ ôóíêö³é<br />
â åêîíîì³ö³). Ââåäåìî ïîçíà÷åííÿ:<br />
Q K<br />
q = , k<br />
L<br />
= L<br />
. (11.9.13)<br />
Óâåäåí³ âåëè÷èíè â³äïîâ³äíî íàçèâàþòüñÿ ñåðåäíüîþ ïðîäóêòèâí³ñòþ<br />
ïðàö³ ³ ñåðåäíüîþ ôîíäîîçáðîºí³ñòþ.<br />
Âðàõîâóþ÷è ïîçíà÷åííÿ (11.8.13) ³ îäíîð³äí³ñòü ôóíêö³¿<br />
Q = Q(K, L), îòðèìàºìî<br />
( , )<br />
Q Q K L ⎛K<br />
⎞<br />
q = = = Q ,1 Q( k,1) q( k)<br />
L L<br />
⎜ = =<br />
L<br />
⎟<br />
.<br />
⎝ ⎠<br />
Äàë³, ïðîäèôåðåíöþºìî âåëè÷èíó k ïî çì³íí³é t.  ðåçóëüòàò³<br />
ìàòèìåìî:<br />
(<br />
K<br />
)<br />
dk d<br />
L K′ L − KL′ K′ L′<br />
= = = − K =<br />
2 2<br />
dt dt L L L<br />
ρQ −µ K K<br />
= −ν = ρq( k) − ( µ +ν ) k.<br />
L L<br />
dk<br />
=ρ − µ+ν k, (11.9.14)<br />
dt<br />
Îñòàòî÷íî: q( k) ( )<br />
k<br />
K<br />
0<br />
( 0) k<br />
=<br />
0<br />
=<br />
L<br />
. (11.9.15)<br />
0<br />
Çàóâàæèìî, ùî â ëàíöþæêó ïåðåòâîðåíü, ÿê³ ïðèâåëè íàñ<br />
äî äèôåðåíö³àëüíîãî ð³âíÿííÿ (11.9.4) ³ ïî÷àòêîâî¿ óìîâè<br />
(11.9.15), ìè ñêîðèñòàëèñÿ ñï³ââ³äíîøåííÿìè (11.9.13).<br />
Ïðèêëàä 11.9.3. Íåõàé ( )<br />
ρ= 10 −1 ,<br />
1 1<br />
, 10 9 2 2<br />
Q K L = ⋅ K L , k<br />
0<br />
= 5 ,<br />
1<br />
µ=ν= . Òðåáà çíàéòè ñåðåäíþ ôîíäîîçáðîºí³ñòü<br />
2<br />
ï³äïðèºìñòâà.<br />
гâíÿííÿ (11.9.14) ïðè çàäàíèõ óìîâàõ íàáóäå âèãëÿäó:<br />
dk<br />
k k<br />
dt + = ⋅<br />
1<br />
10 5 2<br />
448 449
Change language