ゲーム理論BASIC 演習63 -最適賃金設定-
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The document discusses solving an optimization problem to find the maximum value of a function w subject to constraints. It is determined that the fixed point wF is 3c and the sliding point wS is 48/13c, which is greater than the fixed point. Tables are presented showing the variables and values at the fixed and sliding points.
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ゲーム理論BASIC 演習63 -最適賃金設定-
- 2. ղ ղ
- 3. ༻ޏओ ਓͷ࿑ಇऀ͘͠ਓͷ࿑ಇऀͱܖΛ͠ ͋ΔλεΫͷ࣮ߦΛߟ͍͑ͯΔ ਓͷ࿑ಇऀΛͨͬޏ߹ʹ͍ͭͯ λεΫͷޭ֬ ࿑ಇऀ͕ྗͨ͠߹ Ͱ ྗ͠ͳ͍߹ Ͱ͋Δͱ͢Δ ྗͨ͠߹ʹ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ λεΫ͕ޭͨ͠ͱ͖ ༻ޏओ ͷརӹΛಘͯ ࿑ಇऀۚ ΛಘΔࣦഊͷ߹ʹ͍ޓԿಘͳ͍ͱ͢Δ ·ͨ ͦͦܖΛ݁ͳ͍߹ʹ͍ޓԿಘͳ͍ͱ͢Δ ༻ޏओ͓Αͼ࿑ಇऀʹ͍ͭͯϦεΫதཱతͳޮ༻ؔ ΛͭͱԾఆ͢Δ ྗ͢Δ͜ͱΛ͖Ͱ੍ڧΔ߹ͷ࠷దͳۚ ΛٻΊΑ ࢀՃ੍ͷΈߟ͑ͨ࠷దԽΛղ͚ྑ͍ ྗ͢Δ͜ͱΛ͍ͳ͖Ͱ੍ڧ߹ͷ࠷దͳۚ ΛٻΊΑ ࢀՃ੍͓Αͼ༠Ҿཱ྆ੑ݅ ࿑ಇऀʹͱͬͯྗ͢Δํ͕ ྗ͠ͳ͍ΑΓಘ͢ΔͨΊͷ݅ Λߟ͑ͯ ࠷దԽΛղ͚ྑ͍ˎྗͤͨ͞ํ͕༻ޏओʹͱͬͯಘͰ͋Δͱͯ͠ྑ͍ #4*$ԋशࢀর 1 3 1 16 c 0 R 0 w u(x) = x wF wS
- 4. ղ λεΫͷޭ֬࿑ಇऀ͕ྗͨ͠߹ Ͱ ྗ͠ͳ͍߹ Ͱ͋Δͱ͢Δ ྗͨ͠߹ʹ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ λεΫ͕ޭͨ͠ͱ͖ ༻ޏओ ͷརӹΛಘͯ ࿑ಇऀۚ ΛಘΔࣦഊͷ߹ʹ͍ޓԿಘͳ͍ͱ͢Δ ༻ޏओ͓Αͼ࿑ಇऀʹ͍ͭͯϦεΫதཱతͳޮ༻ؔ ΛͭͱԾఆ͢Δ ྗ͢Δ͜ͱΛ͖Ͱ੍ڧΔ߹ͷ࠷దͳۚ ΛٻΊΑࢀՃ੍ͷΈߟ͑ͨ࠷దԽΛղ͚ྑ͍ TU ੍݅ ΑΓ త͕ؔ࠷େͱͳΔͷ Ώ͑ʹ࠷దͳۚ ͱͳΔ 1 3 1 16 c 0 R 0 w u(x) = x wF max w 1 3 (R − w) 1 3 (w − c) + 2 3 (−c) ≥ 0 1 3 (w − c) + 2 3 (−c) ≥ 0 ⇔ w ≥ 3c w = 3c wF = 3c w ࢀՃ ڋ൱ ྗ͢Δ ྗ͠ͳ͍ ޭ ࣦഊ ޭ ࣦഊ ༻ޏओ ࿑ಇऀ ࣗવ R − w w − c D ڋ൱ ڋ൱
- 5. ղ λεΫͷޭ֬࿑ಇऀ͕ྗͨ͠߹ Ͱ ྗ͠ͳ͍߹ Ͱ͋Δͱ͢Δ ྗͨ͠߹ʹ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ λεΫ͕ޭͨ͠ͱ͖ ༻ޏओ ͷརӹΛಘͯ ࿑ಇऀۚ ΛಘΔࣦഊͷ߹ʹ͍ޓԿಘͳ͍ͱ͢Δ ༻ޏओ͓Αͼ࿑ಇऀʹ͍ͭͯϦεΫதཱతͳޮ༻ؔ ΛͭͱԾఆ͢Δ ྗ͢Δ͜ͱΛ͍ͳ͖Ͱ੍ڧ߹ͷ࠷దͳۚ ΛٻΊΑ TU ੍݅ ͔ͭ త͕ؔ࠷େͱͳΔͷ Ώ͑ʹ࠷దͳۚ ͱͳΔ 1 3 1 16 c 0 R 0 w u(x) = x wS max w 1 3 (R − w) 1 3 (w − c) + 2 3 (−c) ≥ 0 1 3 (w − c) + 2 3 (−c) ≥ 1 16 w + 15 16 0 1 3 (w − c) + 2 3 (−c) ≥ 0 ⇔ w ≥ 3c 1 3 (w − c) + 2 3 (−c) ≥ 1 16 w + 15 16 0 ⇔ w ≥ 48 13 c w = 48 13 c wS = 48 13 c( 3c = wF ) w ࢀՃ ڋ൱ ྗ͢Δ ྗ͠ͳ͍ ޭ ࣦഊ ޭ ࣦഊ ༻ޏओ ࿑ಇऀ ࣗવ R − w w − c D R − w w ༠Ҿཱ྆ੑ݅ ༠Ҿཱ྆ੑ݅ͷͱͰ ྗ͢ΔΛબͤΔ࣌ͷظརಘ
- 6. ༻ޏओ ਓͷ࿑ಇऀ͘͠ਓͷ࿑ಇऀͱܖΛ͠ ͋ΔλεΫͷ࣮ߦΛߟ͍͑ͯΔ ਓͷ࿑ಇऀΛͨͬޏ߹ʹ͍ͭͯ λεΫͷޭ֬ ӈදͷΑ͏ʹͳΔͱ͢Δ ྗͨ͠߹ʹ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ λεΫ͕ޭͨ͠ͱ͖ ༻ޏओ ͷརӹΛಘͯ ਓͷ࿑ಇऀͦΕͧΕۚ ΛಘΔ ࣦഊͷ߹શһԿಘͳ͍ͱ͢Δ·ͨ ͦͦܖΛ݁ͳ͍߹ʹ͍ޓԿಘͳ͍ͱ͢Δ ༻ޏओ͓Αͼ࿑ಇऀʹ͍ͭͯϦεΫதཱతͳޮ༻ؔ ΛͭͱԾఆ͢Δ ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͖Ͱ੍ڧΔ߹ʹ࠷దͳۚ ΛٻΊΑ ࢀՃ੍Λߟ͑ͯ ࠷దԽΛղ͚ྑ͍ ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͍ͳ͖Ͱ੍ڧ߹ʹ࠷దͳۚ ΛٻΊΑ ࢀՃ੍͓Αͼ༠Ҿཱ྆ੑ݅ ࿑ಇऀʹͱͬͯྗ͢Δํ͕ ྗ͠ͳ͍ΑΓಘ͢ΔͨΊͷ݅ Λߟ͑ͯ ࠷దԽΛղ͚ྑ͍ˎྗͤͨ͞ํ͕༻ޏओʹͱͬͯಘͰ͋Δͱͯ͠ྑ͍ c 0 R 0 w′  u(x) = x w′  F w′  S ࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ ྗ͢Δ
- 7. ࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ ྗ͢Δ w′  ࢀՃ ڋ൱ ྗ͢Δ ྗ͠ͳ͍ ༻ޏओ ࿑ಇऀ ࣗવ R − 2w′  w′  − c w′  − c 0 −c −c R − 2w′  w′  − c w′  ࿑ಇऀ R − 2w′  w′  w′  − c 0 0 −c 0 0 0 R − 2w′  w′  w′  ྗ͢Δ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ 0 −c 0
- 8. ղ ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͖Ͱ੍ڧΔ߹ʹ࠷దͳۚ ΛٻΊΑ ࢀՃ੍Λߟ͑ͯ ࠷దԽΛղ͚ྑ͍ TU త͕ؔ࠷େͱͳΔͷ w′  F max w′  2 3 (R − 2w′  ) 2 3 (w′  − c) + 1 3 (−c) ≥ 0 w′  F = 3 2 c ࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ ྗ͢Δ w′  ࢀՃ ڋ൱ ྗ͢Δ ྗ͠ͳ͍ ༻ޏओ ࿑ಇऀ ࣗવ R − 2w′  w′  − c w′  − c 0 −c −c R − 2w′  w′  w′  ࿑ಇऀ R − 2w′  w′  w′  0 0 −c 0 0 0 R − 2w′  w′  w′  ྗ͢Δ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ 0 −c 0
- 9. ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͍ͳ͖Ͱ੍ڧ߹ʹ࠷దͳۚ ΛٻΊΑ ˎਓͷ࿑ಇऀʹྗͤͨ͞ํ͕༻ޏओʹͱͬͯಘͰ͋Δͱͯ͠ྑ͍ ૬खͷͯ͢ߦಈʹରͯ͠ ʮྗ͢Δʯ͕࠷దͱͳΔ݅Λߟ͑Δ ࿑ಇऀͷʮྗ͠ͳ͍ʯʹର͠ ྗ͢Δͷ͕࠷దʹͳΔͷ ࿑ಇऀͷʮྗ͢Δʯʹର͠ ྗ͢Δͷ͕࠷దʹͳΔͷ Ώ͑ʹ ͷ͕݅ྗ͢ΔͨΊͷ݅Ͱ͋Δ w′  S 1 8 w′  ≤ 1 3 w′  − c ⇔ w′  ≥ 24 5 c 1 3 w′  ≤ 2 3 w′  − c ⇔ w′  ≥ 3c w′  ≥ 24 5 c ղ ࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ ྗ͢Δ w ࢀՃ ڋ൱ ྗ͢Δ ྗ͠ͳ͍ ༻ޏओ ࿑ಇऀ ࣗવ R − 2w′  w′  − c w′  − c 0 −c −c R − 2w′  w′  − c w′  ࿑ಇऀ R − 2w′  w′  w′  − c 0 0 −c 0 0 0 R − 2w′  w′  w′  ྗ͢Δ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ 0 −c 0 ࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ X` X` ྗ͢Δ X`D X`D ͍͑ݴΕ ʮྗ͢Δʯ͕ ऑ ࢧઓུͱͳΔͨΊͷ݅
- 10. ղ ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͍ͳ͖Ͱ੍ڧ߹ʹ࠷దͳۚ ΛٻΊΑ ˎਓͷ࿑ಇऀʹྗͤͨ͞ํ͕༻ޏओʹͱͬͯಘͰ͋Δͱͯ͠ྑ͍ TU త͕ؔ࠷େͱͳΔͷ w′  S max w′  2 3 (R − 2w′  ) 2 3 (w′  − c) + 1 3 (−c) ≥ 0 ⇔ w′  ≥ 3 2 c w′  ≥ 24 5 c w′  S = 24 5 c ࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ ྗ͢Δ ࢀՃ ڋ൱ ྗ͢Δ ྗ͠ͳ͍ ࿑ಇऀ ࣗવ R − 2w′  w′  − c w′  − c 0 −c −c R − 2w′  w′  − c w′  ࿑ಇऀ R − 2w′  w′  w′  − c 0 0 −c 0 0 0 R − 2w′  w′  w′  ྗ͢Δ ྗ͠ͳ͍ ྗ͢Δ ྗ͠ͳ͍ 0 −c 0 w′  3 2 c = 15 10 c 3c = 30 10 c 24 5 c = 48 10 c ૬ख͕ྗ͠ͳͯ͘ྗ͢Δ ૬ख͕ྗ͠ͳ͍ͷͰ͋Ε ྗ͠ͳ͍ ࢀՃ੍