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äàòêîâ³é âàðòîñò³ ïðîäóêö³¿ çà ðàõóíîê çá³ëüøåííÿ ôîíä³â<br />
ï³äïðèºìñòâà íà îäíó îäèíèöþ (êóïèëè îäèí âåðñòàê).<br />
×àñòèííà ïîõ³äíà â³ä âèðîáíè÷î¿ ôóíêö³¿ z = f(x, y) ïî<br />
îáñÿãó òðóäîâèõ ðåñóðñ³â â òî÷ö³ M 0 (x 0 , y 0 ) íàáëèæåíî äîð³âíþº<br />
äîäàòêîâ³é âàðòîñò³ ïðîäóêö³¿, âèðîáëåíî¿ ùå îäíèì<br />
äîäàòêîâèì ðîá³òíèêîì.<br />
z′ x , y òà<br />
y<br />
Ââåäåí³ òóò ÷àñòèíí³ ïîõ³äí³ â òî÷ö³ x ( 0 0)<br />
( 0,<br />
0)<br />
z′ x y íàçèâàþòüñÿ â³äïîâ³äíî ãðàíè÷íîþ ïðîäóêòèâí³ñòþ<br />
òà ãðàíè÷íîþ ôîíäîâ³ääà÷åþ.<br />
×åðåç ò³æ ïðè÷èíè, ùî áóëè ïîäàí³ ðàí³øå, â åêîíîì³ö³<br />
(äèâ. ï. 7), ââîäÿòü òàêå âàæëèâå ïîíÿòòÿ, ÿê ÷àñòèííà åëàñòè÷í³ñòü.<br />
Äëÿ ðîçãëÿíóòî¿ âèðîáíè÷î¿ ôóíêö³¿ z = f(x, y)<br />
÷àñòèíí³ åëàñòè÷íîñò³ â òî÷ö³ M 0 (x 0 , y 0 ) ìàþòü âèãëÿä:<br />
z x<br />
E ( , ) 0<br />
x<br />
x y = z′<br />
( x , y ), (10.3.10)<br />
0 0 x 0 0<br />
z0<br />
z y<br />
E ( , ) 0<br />
y<br />
x y = z′<br />
( x , y ), (10.3.11)<br />
0 0 y 0 0<br />
z0<br />
äå z 0 = f(x 0 , y 0 ).<br />
Ïðèêëàä 10.3.2. Íåõàé âèðîáíèöòâî äåÿêîãî ï³äïðèºìñòâà<br />
õàðàêòåðèçóºòüñÿ âèðîáíè÷îþ ôóíêö³ºþ Êîááà – Äóãëàñà<br />
z = Ax α y β , α +β≤1, x — îáñÿã ôîíä³â, y — îáñÿã òðóäîâèõ ðåñóðñ³â<br />
(íàïðèêëàä, ê³ëüê³ñòü ðîá³òíèê³â).<br />
Òðåáà çíàéòè ãðàíè÷íó ïðîäóêòèâí³ñòü ïðàö³ òà ãðàíè÷íó<br />
ôîíäîâ³ääà÷ó ïðè x =25⋅ 10 8 , y = 1000.<br />
Ðîçâ’ÿçàííÿ<br />
α−1<br />
β<br />
1<br />
z′ = Aα⋅ x y , z′ = Aβ⋅ x α y<br />
β− ;<br />
x<br />
y<br />
( 8 ) ( 8 α−<br />
A ) 1<br />
β<br />
z′ 25 ⋅ 10 ,1000 = α⋅ 25 ⋅ 10 1000 ,<br />
x<br />
( ) A ( )<br />
8 8 α<br />
β− 1<br />
z′ 25 ⋅ 10 ,1000 = β⋅ 25 ⋅ 10 1000 .<br />
y<br />
Ïðèêëàä 10.3.3. Íåõàé ñïðàâäæóþòüñÿ óìîâè ïðèêëàäó<br />
10.3.2.<br />
Òðåáà âèçíà÷èòè ïàðàìåòðè A, α ³ β âèðîáíè÷î¿ ôóíêö³¿<br />
Êîááà – Äóãëàñà, ÿêùî â³äîìî, ùî äëÿ çá³ëüøåííÿ âèïóñêó<br />
ïðîäóêö³¿ íà 5% íåîáõ³äíî çá³ëüøèòè ôîíäè íà 10% àáî<br />
çá³ëüøèòè ÷èñåëüí³ñòü ðîá³òíèê³â íà 15%.  ìîìåíò äîñë³äæåííÿ<br />
îäèí ðîá³òíèê çà ì³ñÿöü âèðîáëÿº ïðîäóêö³þ íà<br />
100 000 ãðí., à âñüîãî ðîá³òíèê³â 1000.<br />
Ðîçâ’ÿçàííÿ. Çíàõîäèìî ñïî÷àòêó ÷àñòèíí³ åëàñòè÷íîñò³<br />
z<br />
x<br />
E ( 25 ⋅ 10 8 ,10<br />
3 ) = α− 1 β<br />
x<br />
A x y<br />
α β<br />
8 8<br />
Ax y x = 25 ⋅ 10 ⋅ α x = 25 ⋅10<br />
= α .<br />
3 3<br />
y = 10 y = 10<br />
5 1<br />
Ç ³íøîãî áîêó, ³ç óìîâ çàäà÷³ âèïëèâàº, ùî α= = .<br />
10 2<br />
z<br />
8 3 5 1<br />
Àíàëîã³÷íî ïîêàçóºòüñÿ, ùî Ey<br />
( 25⋅ 10 ,10 ) =β= = .<br />
15 3<br />
Òàêèì ÷èíîì, åëàñòè÷í³ñòü β âèïóñêó ïðîäóêö³¿ ïî òðóäó<br />
äîð³âíþº 1 3 , à åëàñòè÷í³ñòü α ïî ôîíäàõ äîð³âíþº 1 2 .<br />
Âðàõîâóþ÷è öåé ôàêò, âèðîáíè÷ó ôóíêö³þ Êîááà – Äóãëàñà<br />
çàïèñóºìî ó âèãëÿä³:<br />
1 1<br />
2 3<br />
.<br />
z = Ax y<br />
ϳäñòàâèìî ³íø³ äàíí³ çàäà÷³ ³ îòðèìàºìî ð³âí³ñòü<br />
1 1<br />
8 2 3<br />
4 5<br />
( ) ( )<br />
100000 ⋅ 1000 = A ⋅ 25 ⋅10 ⋅ 1000 = A⋅5 ⋅10 ⋅ 10 = A ⋅5 ⋅ 10 ,<br />
çâ³äêè, A = 200.<br />
Îòæå, îñòàòî÷íî ôóíêö³ÿ Êîááà – Äóãëàñà ìຠâèãëÿä<br />
1 1<br />
2 3<br />
z = 200x y .<br />
Çàóâàæåííÿ. Äëÿ çðó÷íîñò³ ïðè ðîçâ’ÿçóâàíí³ ïðèêëàäó<br />
ìè çàì³íèëè òðàäèö³éí³ çì³íí³ K, L ³ y íà x, y ³ z.<br />
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