Стоимость в экономических системах. Светлов Н.М. - 27 стр.

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‰Îfl ÍÓÚÓðÓÈ ‚˚ÔÓÎÌfl˛ÚÒfl ÛÒÎÓ‚Ëfl (2.4), ̇Á˚‚‡˛Ú ÚÓ˜ÍÓÈ äÛ̇-
í‡ÍÍÂð‡.
ÖÒÎË ‚ÂÍÚÓð x* Ôð‰ÒÚ‡‚ÎflÂÚ ÒÓ·ÓÈ ÓÔÚËχθÌÓ ð¯ÂÌË Á‡‰‡-
˜Ë (2.1), ÚÓ ÍÓÏÔÓÌÂÌÚ˚ ‚ÂÍÚÓð‡ л*, ‚ Ô‡ðÂ Ò ÍÓÚÓð˚Ï x* Ó·ð‡ÁÛÂÚ ÚÓ˜-
ÍÛ äÛ̇-í‡ÍÍÂð‡, ð‡‚Ì˚ ‚Â΢ËÌ‡Ï ÔðËðÓÒÚ‡ F(x) ‚ÒΉÒÚ‚Ë ۂÂÎË-
˜ÂÌËfl ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘Ëı ÍÓÏÔÓÌÂÌÚÓ‚ ‚ÂÍÚÓð‡ y ̇ ·ÂÒÍÓ̘ÌÓ Ï‡ÎÛ˛.
ùÚË ‚Â΢ËÌ˚ ËÁÏÂðfl˛ÚÒfl ‚ ‰ËÌˈ‡ı F(x) ‚ ð‡Ò˜fiÚ ̇ ‰ËÌËˆÛ ËÁ-
ÏÂðÂÌËfl ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘Â„Ó Ó„ð‡Ì˘ÂÌËfl. èðËÏÂÌËÚÂθÌÓ Í ÒËÒÚÂÏ E
Ï˚ ̇Á‚‡ÎË ˝ÚË ‚Â΢ËÌ˚ Á̇˜ÂÌËflÏË ÒÚÓËÏÓÒÚË.
í‡ÍËÏ Ó·ð‡ÁÓÏ, ‚ ÔðÓˆÂÒÒ ð¯ÂÌËfl Á‡‰‡˜Ë (2.1), ÓÔËÒ˚‚‡˛˘ÂÈ
ÒËÒÚÂÏÛ E, ̇ðfl‰Û Ò ÓÔÚËχθÌ˚Ï ‚ÂÍÚÓðÓÏ x* ‚˚˜ËÒÎ
flÂÚÒfl ‚ÂÍÚÓð л*
ÒÚÓËÏÓÒÚË ðÂÒÛðÒÓ‚.
óÂÚ‚fiðÚÓ ÛÒÎÓ‚Ë äÛ̇-í‡ÍÍÂð‡ (ÔÓÒÍÓθÍÛ
∂L
∂
λ
i
= f
i
(x)) ÏÓÊÌÓ
ÔÂðÂÔËÒ‡Ú¸ ‚ ÙÓðÏÂ
λ
i
* f
i
(x*) = 0. Ç ˝ÚÓÈ ÙÓðÏ ÓÌÓ ËÁ‚ÂÒÚÌÓ ÔÓ‰ ̇-
Á‚‡ÌËÂÏ ÛÒÎÓ‚Ëfl ‰ÓÔÓÎÌfl˛˘ÂÈ ÌÂÊfiÒÚÍÓÒÚË. éÌÓ „‡ð‡ÌÚËðÛÂÚ, ˜ÚÓ
λ
i
* ≠ 0 ÚÓθÍÓ ÚÓ„‰‡, ÍÓ„‰‡ f
i
(x*) = 0, Ú.Â. ÍÓ„‰‡ ðÂÒÛðÒ i ËÒÔÓθÁÛÂÚÒfl
ÔÓÎÌÓÒÚ¸˛. àÁ ÛÒÎÓ‚Ëfl ‰ÓÔÓÎÌfl˛˘ÂÈ ÌÂÊfiÒÚÍÓÒÚË ÒΉÛÂÚ, ˜ÚÓ ‚ ÚÓ˜ÍÂ
äÛ̇-í‡ÍÍÂð‡ ÚÓθÍÓ Ó‰ËÌ ˜ÎÂÌ ÙÛÌ͈ËË ã‡„ð‡Ìʇ ÏÓÊÂÚ ÓÚ΢‡Ú¸Òfl
ÓÚ ÌÛÎfl, ‡ ËÏÂÌÌÓ F(x). ëΉӂ‡ÚÂθÌÓ, ‚ ÚӘ͠äÛ̇-í‡ÍÍÂð‡, ÒÓÓÚ-
‚ÂÚÒÚ‚Û˛˘ÂÈ ÓÔÚËÏÛÏÛ Á‡‰‡˜Ë (2.1), Á̇˜ÂÌË ÙÛÌ͈ËË ã‡„ð‡Ìʇ
ð‡‚ÌÓ ÓÔÚËχθÌÓÏÛ Á̇˜ÂÌ˲ ÙÛÌ͈ËË Ôð‰ÔÓ˜ÚÂÌËfl ÒËÒÚÂÏ˚ E.
èð‰ÒÚ‡‚ËÏ ÙÛÌÍˆË˛ ㇄ð‡Ìʇ ‚ ÙÓðÏÂ
L(x, л) = F(x) –
∑
i∈I
λ
i
q
i
(x) +
∑
i∈I
λ
i
y
i
. (2.5)
ä‡Ê‰˚È ˜ÎÂÌ л
i
q
i
(x) ‚˚ð‡ÊÂÌËfl (2.5) Ôð‰ÒÚ‡‚ÎflÂÚ ÒÓ·ÓÈ ÒÚÓËÏÓÒÚ¸
ÔÓÚð·ÎflÂÏÓ„Ó ÍÓ΢ÂÒÚ‚‡ ðÂÒÛðÒ‡ i, ‡ л
i
y
i
— ÒÚÓËÏÓÒÚ¸ ÍÓ΢ÂÒÚ‚‡
ðÂÒÛðÒ‡ i, ÔÓÒÚÛÔ‡˛˘Â„Ó ‚ ÏÓ‰ÂÎËðÛÂÏÛ˛ ÒËÒÚÂÏÛ. àÁ ˜ÂÚ‚fiðÚÓ„Ó ÛÒ-
ÎÓ‚Ëfl äÛ̇-í‡ÍÍÂð‡ ÒΉÛÂÚ, ˜ÚÓ ‚ β·ÓÈ ÚӘ͠äÛ̇-í‡ÍÍÂð‡ ‰Îfl
β·Ó„Ó ·Î‡„‡ i, Í‡Í ‰ÂÙˈËÚÌÓ„Ó, Ú‡Í Ë ÌÂÓ„ð‡Ì˘ÂÌÌÓ„Ó, ËÏÂÂÚ ÏÂÒÚÓ
л
i
q
i
(x) = л
i
y
i
. ëÏ˚ÒÎ ˝ÚÓ„Ó ð‡‚ÂÌÒÚ‚‡ ‚ ÚÓÏ, ˜ÚÓ ‚Òfl ÒÚÓËÏÓÒÚ¸, ÔðËÓ·-
ðÂÚ‡Âχfl ‚ ÙÓðÏ ·Î‡„, ÔÓÒÚÛÔ‡˛˘Ëı ËÁ Òð‰˚ ÒËÒÚÂÏ˚ E, ð‡ÒıÓ‰ÛÂÚ-
Òfl ‚ ÔðÓˆÂÒÒ Ëı ÔÓÚð·ÎÂÌËfl (ÔðÂÓ·ð‡ÁÓ‚‡ÌËfl ‚ ˆÂ΂Ӡ·Î‡„Ó),
‰Îfl ÍÓÚÓðÓÈ ‚˚ÔÓÎÌfl˛ÚÒfl ÛÒÎÓ‚Ëfl (2.4), ̇Á˚‚‡˛Ú ÚÓ˜ÍÓÈ äÛ̇-
í‡ÍÍÂð‡.
       ÖÒÎË ‚ÂÍÚÓð x* Ôð‰ÒÚ‡‚ÎflÂÚ ÒÓ·ÓÈ ÓÔÚËχθÌÓ ð¯ÂÌË Á‡‰‡-
˜Ë (2.1), ÚÓ ÍÓÏÔÓÌÂÌÚ˚ ‚ÂÍÚÓð‡ л*, ‚ Ô‡ðÂ Ò ÍÓÚÓð˚Ï x* Ó·ð‡ÁÛÂÚ ÚÓ˜-
ÍÛ äÛ̇-í‡ÍÍÂð‡, ð‡‚Ì˚ ‚Â΢ËÌ‡Ï ÔðËðÓÒÚ‡ F(x) ‚ÒΉÒÚ‚Ë ۂÂÎË-
˜ÂÌËfl ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘Ëı ÍÓÏÔÓÌÂÌÚÓ‚ ‚ÂÍÚÓð‡ y ̇ ·ÂÒÍÓ̘ÌÓ Ï‡ÎÛ˛.
ùÚË ‚Â΢ËÌ˚ ËÁÏÂðfl˛ÚÒfl ‚ ‰ËÌˈ‡ı F(x) ‚ ð‡Ò˜fiÚ ̇ ‰ËÌËˆÛ ËÁ-
ÏÂðÂÌËfl ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘Â„Ó Ó„ð‡Ì˘ÂÌËfl. èðËÏÂÌËÚÂθÌÓ Í ÒËÒÚÂÏ E
Ï˚ ̇Á‚‡ÎË ˝ÚË ‚Â΢ËÌ˚ Á̇˜ÂÌËflÏË ÒÚÓËÏÓÒÚË.
       í‡ÍËÏ Ó·ð‡ÁÓÏ, ‚ ÔðÓˆÂÒÒ ð¯ÂÌËfl Á‡‰‡˜Ë (2.1), ÓÔËÒ˚‚‡˛˘ÂÈ
ÒËÒÚÂÏÛ E, ̇ðfl‰Û Ò ÓÔÚËχθÌ˚Ï ‚ÂÍÚÓðÓÏ x* ‚˚˜ËÒÎflÂÚÒfl ‚ÂÍÚÓð л*
ÒÚÓËÏÓÒÚË ðÂÒÛðÒÓ‚.
                                                   ∂L
       óÂÚ‚fiðÚÓ ÛÒÎÓ‚Ë äÛ̇-í‡ÍÍÂð‡ (ÔÓÒÍÓθÍÛ      = fi (x)) ÏÓÊÌÓ
                                                   ∂λi
ÔÂðÂÔËÒ‡Ú¸ ‚ ÙÓðÏ λ*i fi (x*) = 0. Ç ˝ÚÓÈ ÙÓðÏ ÓÌÓ ËÁ‚ÂÒÚÌÓ ÔÓ‰ ̇-
Á‚‡ÌËÂÏ ÛÒÎÓ‚Ëfl ‰ÓÔÓÎÌfl˛˘ÂÈ ÌÂÊfiÒÚÍÓÒÚË. éÌÓ „‡ð‡ÌÚËðÛÂÚ, ˜ÚÓ
λ*i ≠ 0 ÚÓθÍÓ ÚÓ„‰‡, ÍÓ„‰‡ fi (x*) = 0, Ú.Â. ÍÓ„‰‡ ðÂÒÛðÒ i ËÒÔÓθÁÛÂÚÒfl
ÔÓÎÌÓÒÚ¸˛. àÁ ÛÒÎÓ‚Ëfl ‰ÓÔÓÎÌfl˛˘ÂÈ ÌÂÊfiÒÚÍÓÒÚË ÒΉÛÂÚ, ˜ÚÓ ‚ ÚÓ˜ÍÂ
äÛ̇-í‡ÍÍÂð‡ ÚÓθÍÓ Ó‰ËÌ ˜ÎÂÌ ÙÛÌ͈ËË ã‡„ð‡Ìʇ ÏÓÊÂÚ ÓÚ΢‡Ú¸Òfl
ÓÚ ÌÛÎfl, ‡ ËÏÂÌÌÓ F(x). ëΉӂ‡ÚÂθÌÓ, ‚ ÚӘ͠äÛ̇-í‡ÍÍÂð‡, ÒÓÓÚ-
‚ÂÚÒÚ‚Û˛˘ÂÈ ÓÔÚËÏÛÏÛ Á‡‰‡˜Ë (2.1), Á̇˜ÂÌË ÙÛÌ͈ËË ã‡„ð‡Ìʇ
ð‡‚ÌÓ ÓÔÚËχθÌÓÏÛ Á̇˜ÂÌ˲ ÙÛÌ͈ËË Ôð‰ÔÓ˜ÚÂÌËfl ÒËÒÚÂÏ˚ E.
      èð‰ÒÚ‡‚ËÏ ÙÛÌÍˆË˛ ㇄ð‡Ìʇ ‚ ÙÓðÏÂ
                   L(x, л) = F(x) –   ∑λi qi (x) + ∑λi yi.           (2.5)
                                      i∈I         i∈I
ä‡Ê‰˚È ˜ÎÂÌ лi qi (x) ‚˚ð‡ÊÂÌËfl (2.5) Ôð‰ÒÚ‡‚ÎflÂÚ ÒÓ·ÓÈ ÒÚÓËÏÓÒÚ¸
ÔÓÚð·ÎflÂÏÓ„Ó ÍÓ΢ÂÒÚ‚‡ ðÂÒÛðÒ‡ i, ‡ лi yi — ÒÚÓËÏÓÒÚ¸ ÍÓ΢ÂÒÚ‚‡
ðÂÒÛðÒ‡ i, ÔÓÒÚÛÔ‡˛˘Â„Ó ‚ ÏÓ‰ÂÎËðÛÂÏÛ˛ ÒËÒÚÂÏÛ. àÁ ˜ÂÚ‚fiðÚÓ„Ó ÛÒ-
ÎÓ‚Ëfl äÛ̇-í‡ÍÍÂð‡ ÒΉÛÂÚ, ˜ÚÓ ‚ β·ÓÈ ÚӘ͠äÛ̇-í‡ÍÍÂð‡ ‰Îfl
β·Ó„Ó ·Î‡„‡ i, Í‡Í ‰ÂÙˈËÚÌÓ„Ó, Ú‡Í Ë ÌÂÓ„ð‡Ì˘ÂÌÌÓ„Ó, ËÏÂÂÚ ÏÂÒÚÓ
лi qi (x) = лi yi. ëÏ˚ÒÎ ˝ÚÓ„Ó ð‡‚ÂÌÒÚ‚‡ ‚ ÚÓÏ, ˜ÚÓ ‚Òfl ÒÚÓËÏÓÒÚ¸, ÔðËÓ·-
ðÂÚ‡Âχfl ‚ ÙÓðÏ ·Î‡„, ÔÓÒÚÛÔ‡˛˘Ëı ËÁ Òð‰˚ ÒËÒÚÂÏ˚ E, ð‡ÒıÓ‰ÛÂÚ-
Òfl ‚ ÔðÓˆÂÒÒ Ëı ÔÓÚð·ÎÂÌËfl (ÔðÂÓ·ð‡ÁÓ‚‡ÌËfl ‚ ˆÂ΂Ӡ·Î‡„Ó),

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