Стоимость в экономических системах. Светлов Н.М. - 141 стр.

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141
A = (a
j
), B = (b
j
), x
0
— ̇˜‡Î¸ÌÓ ÒÓÒÚÓflÌË ÌÂÈχÌÓ‚ÒÍÓÈ ˝ÍÓÌÓÏËÍË,
A
1
, A
2
, B
1
, B
2
— Ú‡ÍË χÚðˈ˚, ˜ÚÓ
A =
⎝
⎜
⎜
⎛
⎠
⎟
⎟
⎞
A
1
A
2
, B =
⎝
⎜
⎜
⎛
⎠
⎟
⎟
⎞
B
1
B
2
, (A
1
– αB
1
) x = 0, (A
2
– αB
2
) x > 0. (3.56)
íÓ„‰‡ ÒËθ̇fl ÚÂÓðÂχ Ó Ï‡„ËÒÚð‡ÎË ‚ ÏÓ‰ÂÎË ÙÓÌ çÂÈχ̇ ËÏÂÂÚ
ÏÂÒÚÓ ÔðË ‚˚ÔÓÎÌÂÌËË ÒÎÂ‰Û˛˘Ëı ÛÒÎÓ‚ËÈ.
1. èӂ‰ÂÌË ˝ÍÓÌÓÏ˘ÂÒÍÓÈ ÒËÒÚÂÏ˚ ÒΉÛÂÚ ‚ÏÂÌfiÌÌ˚Ï Ôð‰-
ÔÓ˜ÚÂÌËflÏ ‚ˉ‡ max
‹ c, x
t'
›, „‰Â t' = sup(T) — ÔÓÒΉÌËÈ ÏÓÏÂÌÚ ‚ðÂ-
ÏÂÌË ÏÓ‰ÂÎËðÛÂÏÓ„Ó ÔÂðËÓ‰‡,
c . 0.
2. 燘‡Î¸ÌÓ ÒÓÒÚÓflÌË ڇÍÓ‚Ó, ˜ÚÓ
Ax - Bx
0
.
3. åÓÊÌÓ Û͇Á‡Ú¸ Ú‡ÍÓÈ ‚ÂÍÚÓð
p' . 0, ‰Îfl ÍÓÚÓðÓ„Ó
p'A = c – pA, p'B = 0.
4. ëÛ˘ÂÒÚ‚Û˛Ú ‰ÂÈÒÚ‚ËÚÂθÌÓ ËÎË ÍÓÏÔÎÂÍÒÌÓÂ
л ≠ –1 Ë ‚ÂÍÚÓð
z, Ì ÒÓ‰Âðʇ˘ËÈ ÌÛ΂˚ı ÍÓÏÔÓÌÂÌÚÓ‚, Ú‡ÍËÂ, ˜ÚÓ (A
1
– лB
1
)z = 0.
5. Ç ÌÂÍÓÚÓðÓÈ ÒÚðÓÍ χÚðˈ˚
A
1
‚Ò ÍÓ˝ÙÙˈËÂÌÚ˚, ÒÓÓÚ‚ÂÚ-
ÒÚ‚Û˛˘Ë ËÒÔÓθÁÛÂÏ˚Ï ÔðÓˆÂÒÒ‡Ï ‚ ÒÓÒÚÓflÌËË ‰Ë̇Ï˘ÂÒÍÓ„Ó ð‡‚-
ÌÓ‚ÂÒËfl, ÏÓ„ÛÚ ·˚Ú¸ ð‡‚Ì˚ ÌÛβ ÚÓθÍÓ ‚ ÒÎÛ˜‡Â, ÍÓ„‰‡ ·Î‡„Ó, ÒÓÓÚ-
‚ÂÚÒÚ‚Û˛˘Â ˝ÚÓÈ ÒÚðÓÍÂ, Ì ÏÓÊÂÚ ·˚Ú¸ ‚˚ÔÛ˘ÂÌÓ ÌË Ó‰ÌËÏ ÔðÓˆÂÒ-
ÒÓÏ.
ç‡ flÁ˚Í ˝ÍÓÌÓÏËÒÚ‡ ˝ÚÓ ÓÁ̇˜‡ÂÚ ÒÎÂ‰Û˛˘ÂÂ. óÚÓ·˚ ‰Îfl ÌÂÈ-
χÌÓ‚ÒÍÓÈ ÏÓ‰ÂÎË ·˚· ‚Âð̇ ÒËθ̇fl ÚÂÓðÂχ Ó Ï‡„ËÒÚð‡ÎË, Ôð‰-
ÔÓ˜ÚËÚÂθÌÓÒÚ¸ ÚÓÈ ËÎË ËÌÓÈ Úð‡ÂÍÚÓðËË Ôӂ‰ÂÌËfl ˝ÍÓÌÓÏ˘ÂÒÍÓÈ
ÒËÒÚÂÏ˚ ‰ÓÎÊ̇ ÓÔð‰ÂÎflÚ¸Òfl ÚÓθÍÓ Âfi ÍÓ̘ÌÓÈ ÚÓ˜ÍÓÈ. ëÍ·‰˚-
‚‡Ú¸Òfl Ó̇ ‰ÓÎÊ̇ ËÁ ‚Â΢ËÌ, ÔðÓÔÓðˆËÓ̇θÌ˚ı ËÌÚÂÌÒË‚ÌÓÒÚflÏ
ÚÂıÌÓÎӄ˘ÂÒÍËı ÔðÓˆÂÒÒÓ‚ ‚ ˝ÚÓÈ ÚÓ˜ÍÂ. Ç ðÂÁÛθڇÚ Á‡‚Âð¯ÂÌËfl
ÚÂıÌÓÎӄ˘ÂÒÍËı ÔðÓˆÂÒÒÓ‚, ÔðÓÚÂ͇˛˘Ëı ‚ ̇˜‡Î¸ÌÓÈ ÚӘ͠Úð‡ÂÍÚÓ-
ðËË, ‰ÓÎÊ̇ ÔðÓËÁ‚Ó‰ËÚ¸Òfl ÔðÓ‰Û͈Ëfl, ‰ÓÒÚ‡ÚӘ̇fl ‰Îfl
ÌÂωÎÂÌÌÓ„Ó
‚˚ıÓ‰‡ ̇ ͇ÍÛ˛-ÎË·Ó ÚÓ˜ÍÛ Ï‡„ËÒÚð‡ÎË. ìÒÎÓ‚Ë 3 Ù‡ÍÚ˘ÂÒÍË ‰ÓÒ-
Ú‡ÚÓ˜ÌÓ ÊfiÒÚÍÓ Á‡‰‡fiÚ Í·ÒÒ Ôð‰ÔÓ˜ÚÂÌËÈ, ÔðË ÍÓÚÓð˚ı ÔðÓËÒıÓ‰ËÚ
‚˚ıÓ‰ ̇ χ„ËÒÚð‡Î¸. Ç Ò‡ÏÓÏ ÚËÔ˘ÌÓÏ ÒÎÛ˜‡Â, ÍÓ„‰‡ ËÏÂÂÚÒfl ‰ËÌ-
ÒÚ‚ÂÌÌÓ ‰Ë̇Ï˘ÂÒÍÓ ð‡‚ÌÓ‚ÂÒËÂ, ‡ Ûð‡‚ÌÂÌËÂ
p'B = 0 ËÏÂÂÚ Â‰ËÌÒÚ-
‚ÂÌÌÓ ð¯ÂÌËÂ, ÙÛÌ͈Ëfl Ôð‰ÔÓ˜ÚÂÌËfl, ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘‡fl ˝ÚËÏ ÛÒÎÓ-
A = (aj), B = (bj), x0 — ̇˜‡Î¸ÌÓ ÒÓÒÚÓflÌË ÌÂÈχÌÓ‚ÒÍÓÈ ˝ÍÓÌÓÏËÍË,
A1, A2, B1, B2 — Ú‡ÍË χÚðˈ˚, ˜ÚÓ
           ⎛A ⎞        ⎛B ⎞
      A = ⎜⎜ ⎟⎟ , B = ⎜⎜ ⎟⎟ , (A1 – αB1) x = 0, (A2 – αB2) x > 0. (3.56)
              1           1

           ⎝ A2 ⎠      ⎝ B2 ⎠
íÓ„‰‡ ÒËθ̇fl ÚÂÓðÂχ Ó Ï‡„ËÒÚð‡ÎË ‚ ÏÓ‰ÂÎË ÙÓÌ çÂÈχ̇ ËÏÂÂÚ
ÏÂÒÚÓ ÔðË ‚˚ÔÓÎÌÂÌËË ÒÎÂ‰Û˛˘Ëı ÛÒÎÓ‚ËÈ.
       1. èӂ‰ÂÌË ˝ÍÓÌÓÏ˘ÂÒÍÓÈ ÒËÒÚÂÏ˚ ÒΉÛÂÚ ‚ÏÂÌfiÌÌ˚Ï Ôð‰-
ÔÓ˜ÚÂÌËflÏ ‚ˉ‡ max ‹ c, xt' ›, „‰Â t' = sup(T) — ÔÓÒΉÌËÈ ÏÓÏÂÌÚ ‚ðÂ-
ÏÂÌË ÏÓ‰ÂÎËðÛÂÏÓ„Ó ÔÂðËÓ‰‡, c . 0.
       2. 燘‡Î¸ÌÓ ÒÓÒÚÓflÌË ڇÍÓ‚Ó, ˜ÚÓ Ax - Bx0.
       3. åÓÊÌÓ Û͇Á‡Ú¸ Ú‡ÍÓÈ ‚ÂÍÚÓð p' . 0, ‰Îfl ÍÓÚÓðÓ„Ó
p'A = c – pA, p'B = 0.
        4. ëÛ˘ÂÒÚ‚Û˛Ú ‰ÂÈÒÚ‚ËÚÂθÌÓ ËÎË ÍÓÏÔÎÂÍÒÌÓ л ≠ –1 Ë ‚ÂÍÚÓð
z, Ì ÒÓ‰Âðʇ˘ËÈ ÌÛ΂˚ı ÍÓÏÔÓÌÂÌÚÓ‚, Ú‡ÍËÂ, ˜ÚÓ (A1 – лB1)z = 0.
       5. Ç ÌÂÍÓÚÓðÓÈ ÒÚðÓÍ χÚðˈ˚ A1 ‚Ò ÍÓ˝ÙÙˈËÂÌÚ˚, ÒÓÓÚ‚ÂÚ-
ÒÚ‚Û˛˘Ë ËÒÔÓθÁÛÂÏ˚Ï ÔðÓˆÂÒÒ‡Ï ‚ ÒÓÒÚÓflÌËË ‰Ë̇Ï˘ÂÒÍÓ„Ó ð‡‚-
ÌÓ‚ÂÒËfl, ÏÓ„ÛÚ ·˚Ú¸ ð‡‚Ì˚ ÌÛβ ÚÓθÍÓ ‚ ÒÎÛ˜‡Â, ÍÓ„‰‡ ·Î‡„Ó, ÒÓÓÚ-
‚ÂÚÒÚ‚Û˛˘Â ˝ÚÓÈ ÒÚðÓÍÂ, Ì ÏÓÊÂÚ ·˚Ú¸ ‚˚ÔÛ˘ÂÌÓ ÌË Ó‰ÌËÏ ÔðÓˆÂÒ-
ÒÓÏ.
       ç‡ flÁ˚Í ˝ÍÓÌÓÏËÒÚ‡ ˝ÚÓ ÓÁ̇˜‡ÂÚ ÒÎÂ‰Û˛˘ÂÂ. óÚÓ·˚ ‰Îfl ÌÂÈ-
χÌÓ‚ÒÍÓÈ ÏÓ‰ÂÎË ·˚· ‚Âð̇ ÒËθ̇fl ÚÂÓðÂχ Ó Ï‡„ËÒÚð‡ÎË, Ôð‰-
ÔÓ˜ÚËÚÂθÌÓÒÚ¸ ÚÓÈ ËÎË ËÌÓÈ Úð‡ÂÍÚÓðËË Ôӂ‰ÂÌËfl ˝ÍÓÌÓÏ˘ÂÒÍÓÈ
ÒËÒÚÂÏ˚ ‰ÓÎÊ̇ ÓÔð‰ÂÎflÚ¸Òfl ÚÓθÍÓ Âfi ÍÓ̘ÌÓÈ ÚÓ˜ÍÓÈ. ëÍ·‰˚-
‚‡Ú¸Òfl Ó̇ ‰ÓÎÊ̇ ËÁ ‚Â΢ËÌ, ÔðÓÔÓðˆËÓ̇θÌ˚ı ËÌÚÂÌÒË‚ÌÓÒÚflÏ
ÚÂıÌÓÎӄ˘ÂÒÍËı ÔðÓˆÂÒÒÓ‚ ‚ ˝ÚÓÈ ÚÓ˜ÍÂ. Ç ðÂÁÛθڇÚ Á‡‚Âð¯ÂÌËfl
ÚÂıÌÓÎӄ˘ÂÒÍËı ÔðÓˆÂÒÒÓ‚, ÔðÓÚÂ͇˛˘Ëı ‚ ̇˜‡Î¸ÌÓÈ ÚӘ͠Úð‡ÂÍÚÓ-
ðËË, ‰ÓÎÊ̇ ÔðÓËÁ‚Ó‰ËÚ¸Òfl ÔðÓ‰Û͈Ëfl, ‰ÓÒÚ‡ÚӘ̇fl ‰Îfl ÌÂωÎÂÌÌÓ„Ó
‚˚ıÓ‰‡ ̇ ͇ÍÛ˛-ÎË·Ó ÚÓ˜ÍÛ Ï‡„ËÒÚð‡ÎË. ìÒÎÓ‚Ë 3 Ù‡ÍÚ˘ÂÒÍË ‰ÓÒ-
Ú‡ÚÓ˜ÌÓ ÊfiÒÚÍÓ Á‡‰‡fiÚ Í·ÒÒ Ôð‰ÔÓ˜ÚÂÌËÈ, ÔðË ÍÓÚÓð˚ı ÔðÓËÒıÓ‰ËÚ
‚˚ıÓ‰ ̇ χ„ËÒÚð‡Î¸. Ç Ò‡ÏÓÏ ÚËÔ˘ÌÓÏ ÒÎÛ˜‡Â, ÍÓ„‰‡ ËÏÂÂÚÒfl ‰ËÌ-
ÒÚ‚ÂÌÌÓ ‰Ë̇Ï˘ÂÒÍÓ ð‡‚ÌÓ‚ÂÒËÂ, ‡ Ûð‡‚ÌÂÌË p'B = 0 ËÏÂÂÚ Â‰ËÌÒÚ-
‚ÂÌÌÓ ð¯ÂÌËÂ, ÙÛÌ͈Ëfl Ôð‰ÔÓ˜ÚÂÌËfl, ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘‡fl ˝ÚËÏ ÛÒÎÓ-

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