㻌㻌㻌㻌ཷ㦂␒ྕ㻛㻱㼤㼍㼙㼕㼚㼑㼑䇻㼟㻌㻺㼡㼙㼎㼑㼞㻌㻌㻌㻌㻌㻌㻌㻌௧࿴ 㻡 ᖺᗘ㻌 㻌 㻞㻜㻞㻟㻌㻭㼏㼍㼐㼑㼙㼕㼏㻌㼅㼑㼍㼞㻌ᮾி኱Ꮫ኱Ꮫ㝔᪂㡿ᇦ๰ᡂ⛉Ꮫ◊✲⛉㻌㻳㼞㼍㼐㼡㼍㼠㼑㻌㻿㼏㼔㼛㼛㼘㻌㼛㼒㻌㻲㼞㼛㼚㼠㼕㼑㼞㻌㻿㼏㼕㼑㼚㼏㼑㼟㻘㻌㼀㼔㼑㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼀㼛㼗㼥㼛㻌㻌㻙㻙㻙㻌 ᅜ㝿༠ຊᏛᑓᨷ㻌 㻙㻙㻙㻌㻰㼑㼜㼍㼞㼠㼙㼑㼚㼠㻌㼛㼒㻌㻵㼚㼠㼑㼞㼚㼍㼠㼕㼛㼚㼍㼘㻌㻿㼠㼡㼐㼕㼑㼟㻌㻌ಟኈㄢ⛬୍⯡ධヨ㻌 㻌 ධᏛヨ㦂ၥ㢟㻌㻻㼞㼐㼕㼚㼍㼞㼥㻌㻱㼤㼍㼙㼕㼚㼍㼠㼕㼛㼚㻌㼒㼛㼞㻌㻹㼍㼟㼠㼑㼞䇻㼟㻌㻯㼛㼡㼞㼟㼑㻌㻌ᑓ㛛⛉┠㻌 㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㻿㼡㼎㼖㼑㼏㼠㻌㻌௧࿴ 㻠 ᖺ 㻤 ᭶ 㻞㻟 ᪥䠄ⅆ䠅㻌 㻭㼡㼓㼡㼟㼠㻌㻞㻟㻌㻔㼀㼡㼑㻕㻘㻌㻞㻜㻞㻞㻌㻝㻠㻦㻜㻜䠉㻝㻡㻦㻝㻜䠄㻣㻜㻌㼙㼕㼚㼡㼠㼑㼟䠅㻌㻌ὀព஦㡯㻌 㻵㼚㼟㼠㼞㼡㼏㼠㼕㼛㼚㼟㻌㻝㻚ヨ㦂㛤ጞ䛾ྜᅗ䛜䛒䜛䜎䛷䚸䛣䛾ၥ㢟෉Ꮚ䜢㛤䛔䛶䛿䛔䛡䜎䛫䜣䚹㻌㻰㼛㻌㼚㼛㼠㻌㼛㼜㼑㼚㻌㼠㼔㼕㼟㻌㼎㼛㼛㼗㼘㼑㼠㻌㼡㼚㼠㼕㼘㻌㼥㼛㼡㻌㼍㼞㼑㻌㼕㼚㼟㼠㼞㼡㼏㼠㼑㼐㻌㼠㼛㻌㼐㼛㻌㼟㼛㻚㻌㻞㻚ゎ⟅䛻䛿䚸ᚲ䛪㯮Ⰽ㖄➹䠄䜎䛯䛿㯮Ⰽ䝅䝱䞊䝥䝨䞁䝅䝹䠅䜢౑⏝䛧䛺䛥䛔䚹㻌㼁㼟㼑㻌㼍㻌㼎㼘㼍㼏㼗㻌㼜㼑㼚㼏㼕㼘㻌㼛㼞㻌㼍㻌㼎㼘㼍㼏㼗㻌㼙㼑㼏㼔㼍㼚㼕㼏㼍㼘㻌㼜㼑㼚㼏㼕㼘㻌㼠㼛㻌㼣㼞㼕㼠㼑㻌㼥㼛㼡㼞㻌㼍㼚㼟㼣㼑㼞㼟㻚㻌㻟㻚ၥ㢟 㻝䡚ၥ㢟 㻟 䛾 㻟 䛴䛾኱ၥ䛾୰䛛䜙 㻝 䛴䛰䛡䜢㑅ᢥ䛧䛶⟅䛘䛺䛥䛔䚹㻞 䛴௨ୖ䛾኱ၥ䜢ゎ⟅䛧䛯ሙྜ䛻䛿䚸᥇Ⅼ䛥䜜䛺䛔䚹㻌㻯㼔㼛㼛㼟㼑㻌㼛㼚㼘㼥㻌㼛㼚㼑㻌㼛㼒㻌㼠㼔㼑㻌㼒㼛㼘㼘㼛㼣㼕㼚㼓㻌㻟㻌㻼㼞㼛㼎㼘㼑㼙㼟㻌㼍㼚㼐㻌㼍㼚㼟㼣㼑㼞㻌㼍㼘㼘㻌㼠㼔㼑㻌㼝㼡㼑㼟㼠㼕㼛㼚㼟㻌㼒㼛㼞㻌㼠㼔㼍㼠㻌㻼㼞㼛㼎㼘㼑㼙㻚㻌㻵㼒㻌㼥㼛㼡㻌㼍㼚㼟㼣㼑㼞㻌㼙㼛㼞㼑㻌㼠㼔㼍㼚㻌㼛㼚㼑㻌㻼㼞㼛㼎㼘㼑㼙㻘㻌㼥㼛㼡㼞㻌㼍㼚㼟㼣㼑㼞㼟㻌㼣㼕㼘㼘㻌㼚㼛㼠㻌㼎㼑㻌㼓㼕㼢㼑㼚㻌㼍㼚㼥㻌㼟㼏㼛㼞㼑㼟㻚㻌㻠㻚ゎ⟅⏝⣬䞉ⲡ✏⏝⣬䛿䛭䜜䛮䜜䠍ᯛ䛷䛩䚹◚ᦆ䛧䛯ሙྜ䜢㝖䛝䚸ゎ⟅⏝⣬䜒䛧䛟䛿ⲡ✏⏝⣬䜢 㻞 ᯛ௨ୖ㓄ᕸ䛩䜛䛣䛸䛿䛷䛝䜎䛫䜣䚹㻌㼅㼛㼡㻌㼔㼍㼢㼑㻌㼎㼑㼑㼚㻌㼜㼞㼛㼢㼕㼐㼑㼐㻌㻝㻌㼍㼚㼟㼣㼑㼞㻌㼟㼔㼑㼑㼠㻌㼍㼚㼐㻌㻝㻌㼐㼞㼍㼒㼠㻌㼟㼔㼑㼑㼠㻚㻌㼁㼚㼘㼑㼟㼟㻌㼠㼔㼑㼥㻌㼍㼞㼑㻌㼐㼍㼙㼍㼓㼑㼐㻘㻌㼥㼛㼡㻌㼙㼍㼥㻌㼚㼛㼠㻌㼑㼤㼏㼔㼍㼚㼓㼑㻌㼛㼞㻌㼞㼑㼝㼡㼑㼟㼠㻌㼍㼐㼐㼕㼠㼕㼛㼚㼍㼘㻌㼟㼔㼑㼑㼠㼟㻚㻌㻡㻚ゎ⟅⏝⣬䛾ᡤᐃḍ䛻䚸ཷ㦂␒ྕ䞉㑅ᢥ䛧䛯ၥ㢟␒ྕ䠄ၥ㢟 㻝䞉ၥ㢟 㻞䞉ၥ㢟 㻟 䛾䛔䛪䜜䛛䠅䜢ᚲ䛪グධ䛧䛺䛥䛔䚹䜎䛯䚸ၥ㢟෉Ꮚ䞉ⲡ✏⏝⣬䛻䜒ཷ㦂␒ྕ䜢グධ䛧䛺䛥䛔䚹ཷ㦂␒ྕ䞉ၥ㢟␒ྕ䛜グධ䛥䜜䛶䛔䛺䛔ሙྜ䛻䛿᥇Ⅼ䛥䜜䛺䛔ሙྜ䛜䛒䜚䜎䛩䚹㻌㻵㼚㻌㼠㼔㼑㻌㼐㼑㼟㼕㼓㼚㼍㼠㼑㼐㻌㼎㼘㼍㼚㼗㼟㻌㼛㼚㻌㼠㼔㼑㻌㼍㼚㼟㼣㼑㼞㻌㼟㼔㼑㼑㼠㻘㻌㼒㼕㼘㼘㻌㼕㼚㻌㼥㼛㼡㼞㻌㻱㼤㼍㼙㼕㼚㼑㼑䇻㼟㻌㼚㼡㼙㼎㼑㼞㻌㼍㼚㼐㻌㼠㼔㼑㻌㼚㼡㼙㼎㼑㼞㻌㼛㼒㻌㼥㼛㼡㼞㻌㼝㼡㼑㼟㼠㼕㼛㼚㻌㼏㼔㼛㼕㼏㼑㻌㻔㻼㼞㼛㼎㼘㼑㼙㻌㻝㻌㻙㻌㻟㻕㻚㻌㻲㼕㼘㼘㻌㼕㼚㻌㼥㼛㼡㼞㻌㻱㼤㼍㼙㼕㼚㼑㼑䇻㼟㻌㼚㼡㼙㼎㼑㼞㻌㼕㼚㻌㼠㼔㼑㻌㼐㼑㼟㼕㼓㼚㼍㼠㼑㼐㻌㼎㼘㼍㼚㼗㼟㻌㼛㼚㻌㼠㼔㼑㻌㼝㼡㼑㼟㼠㼕㼛㼚㻌㼎㼛㼛㼗㼘㼑㼠㻌㼍㼚㼐㻌㼠㼔㼑㻌㼐㼞㼍㼒㼠㻌㼟㼔㼑㼑㼠㻌㼍㼟㻌㼣㼑㼘㼘㻚㻌㻵㼒㻌㼠㼔㼛㼟㼑㻌㼍㼞㼑㻌㼚㼛㼠㻌㼒㼕㼘㼘㼑㼐㻌㼕㼚㻌㼜㼞㼛㼜㼑㼞㼘㼥㻘㻌㼥㼛㼡㼞㻌㼍㼚㼟㼣㼑㼞㼟㻌㼙㼍㼥㻌㼚㼛㼠㻌㼎㼑㻌㼟㼏㼛㼞㼑㼐㻚㻌㻢㻚ゎ⟅⏝⣬䛻䚸ゎ⟅䛻㛵ಀ䛾䛺䛔ᩥᏐ䚸グྕ䚸➢ྕ䛺䛹䜢グධ䛧䛶䛿䛔䛡䜎䛫䜣䚹㻌㻰㼛㻌㼚㼛㼠㻌㼙㼍㼞㼗㻌㼛㼞㻌㼙㼍㼗㼑㻌㼍㼚㼥㻌㼕㼞㼞㼑㼘㼑㼢㼍㼚㼠㻌㼟㼥㼙㼎㼛㼘㼟㻌㼛㼞㻌㼣㼞㼕㼠㼕㼚㼓㼟㻌㼛㼚㻌㼥㼛㼡㼞㻌㼍㼚㼟㼣㼑㼞㻌㼟㼔㼑㼑㼠㻚㻌㻣㻚ၥ㢟෉Ꮚ䞉ゎ⟅⏝⣬䞉ⲡ✏⏝⣬䜢ᣢ䛱ᖐ䛳䛶䛿䛔䛡䜎䛫䜣䚹㻌㻰㼛㻌㼚㼛㼠㻌㼠㼍㼗㼑㻌㼠㼔㼑㻌㼝㼡㼑㼟㼠㼕㼛㼚㻌㼎㼛㼛㼗㼘㼑㼠㻘㻌㼍㼚㼟㼣㼑㼞㻌㼟㼔㼑㼑㼠㻘㻌㼍㼚㼐㻌㼐㼞㼍㼒㼠㻌㼟㼔㼑㼑㼠㻌㼛㼡㼠㻌㼛㼒㻌㼠㼔㼑㻌㼞㼛㼛㼙㻚㻌㻤㻚ヨ㦂᫬㛫䛿 㻣㻜 ศ䛷䛩䚹䛯䛰䛧䚸ヨ㦂㛤ጞᚋ 㻟㻜 ศ䜢⤒㐣䛧䛯ᚋ䛿䚸ၥ㢟෉Ꮚ䞉ゎ⟅⏝⣬䞉ⲡ✏⏝⣬䜢ヨ㦂┘╩䛻ᥦฟ䛧䛯䛖䛘䛷䚸㏥ฟ䛧䛶䜒䛛䜎䛔䜎䛫䜣䚹㻌㼀㼔㼑㻌㼑㼤㼍㼙㼕㼚㼍㼠㼕㼛㼚㻌㼠㼕㼙㼑㻌㼕㼟㻌㻣㻜㻌㼙㼕㼚㼡㼠㼑㼟㻚㻌㼅㼛㼡㻌㼙㼡㼟㼠㻌㼞㼑㼙㼍㼕㼚㻌㼕㼚㻌㼠㼔㼑㻌㼞㼛㼛㼙㻌㼒㼛㼞㻌㼍㼠㻌㼘㼑㼍㼟㼠㻌㻟㻜㻌㼙㼕㼚㼡㼠㼑㼟㻌㼛㼚㼏㼑㻌㼠㼔㼑㻌㼑㼤㼍㼙㼕㼚㼍㼠㼕㼛㼚㻌㼟㼠㼍㼞㼠㼟㻚㻌 㻌㼃㼔㼑㼚㻌㼥㼛㼡㻌㼘㼑㼍㼢㼑㻘㻌㼥㼛㼡㻌㼙㼡㼟㼠㻌㼟㼡㼎㼙㼕㼠㻌㼥㼛㼡㼞㻌㼝㼡㼑㼟㼠㼕㼛㼚㻌㼎㼛㼛㼗㼘㼑㼠㻘㻌㼍㼚㼟㼣㼑㼞㻌㼟㼔㼑㼑㼠㻘㻌㼍㼚㼐㻌㼐㼞㼍㼒㼠㻌㼟㼔㼑㼑㼠㻌㼠㼛㻌㼠㼔㼑㻌㼜㼞㼛㼏㼠㼛㼞㻚㻌ࡇࡢ࣮࣌ࢪࡣ✵ⓑ࡛ࡍࠋၥ㢟ࡣḟࡢ࣮࣌ࢪ࠿ࡽጞࡲࡾࡲࡍࠋ
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ၥ㢟㸯࠶ࡿ୺యࡢ⾜ືࡀᕷሙࢆ௓ࡉࡎ࡟௚ࡢ୺య࡟ᙳ㡪ࢆ୚࠼ࡿࡇ࡜ࢆࠕእ㒊ᛶࠖࡲࡓࡣࠕእ㒊⤒῭ࠖ࡜࿧ࡪࠋእ㒊ᛶ࡟ࡣࣉࣛࢫࡢᙳ㡪ࢆ୚࠼ࡿࡶࡢ࡜࣐࢖ࢼࢫࡢᙳ㡪ࢆ୚࠼ࡿࡶࡢࡀᏑᅾࡍࡿࡀࠊᚋ⪅ࡣࠕᕷሙࡢኻᩋࠖࡢせᅉࡢ୍ࡘ࡟ࡶ࡞ࡿࠋၥ㸯㸬ྠࡌศ㔝ࡢ㞽⣽௻ᴗࡀ࠶ࡿ≉ᐃࡢᆅᇦ࡟㞟୰ࡋ࡚❧ᆅࡍࡿ㏵ୖᅜࡢ⏘ᴗ㞟✚ࢆ⪃࠼ࡿࠋ౛࠼ࡤࠊ⦭〇ᕤሙࡸᐙල⏘ᴗࠊ㔠ᒓ㒊ရࡢ㞟✚࡛࠶ࡿࠋ⏘ᴗ㞟✚ࡢ౛ࢆ୍ࡘᣲࡆࠊࡇࡢ⏘ᴗ㞟✚࡛࡝ࡢࡼ࠺࡞ࣉࣛࢫࡢእ㒊ᛶ࡜࣐࢖ࢼࢫࡢእ㒊ᛶࡀ⏕ࡌࡿ࡜⪃࠼ࡽࢀࡿ࠿ࢆㄝ᫂ࡋ࡞ࡉ࠸ࠋၥ㸰㸬እ㒊ᛶࡢゎỴ࡟ࡣࠊእ㒊ᛶࢆࠕෆ㒊໬ࡍࡿࠖࡇ࡜ࡀ᭷⏝ࡔ࡜ࡉࢀ࡚࠸ࡿࡀࠊၥ㸯࡛ᣲࡆࡓ࣐࢖ࢼࢫࡢእ㒊ᛶࢆෆ㒊໬ࡍࡿ௙⤌ࡳࡸ࣓࢝ࢽࢬ࣒ࢆ⪃࠼ࠊㄝ᫂ࡋ࡞ࡉ࠸ࠋ≉࡟ࠊㄡࡀ୺య࡜࡞ࡾࠊ࡝ࡢࡼ࠺࡞⾜ືࢆࡍࡿᚲせࡀ࠶ࡿ࠿ࢆㄽࡌ࡞ࡉ࠸ࠋၥ㸱㸬ၥ㸰࡛ᣲࡆࡓ௙⤌ࡳࡸ࣓࢝ࢽࢬ࣒ࡣࠊ㏵ୖᅜ࡛ࡣୖᡭࡃᶵ⬟ࡋ࡞࠸ࡇ࡜ࡀ࠶ࡿࠋ࡞ࡐࡑࢀࡽࡀୖᡭࡃᶵ⬟ࡋ࡞࠸ྍ⬟ᛶࡀ࠶ࡿ࠿ࢆ⪃ᐹࡋࠊᙜ஦⪅ࢆྲྀࡾᕳࡃᨻ⟇ࡸไᗘ➼ࡢ᮲௳ࢆ࡝ࡢࡼ࠺࡟ᨵၿࡍࢀࡤຠᯝⓗ࡟ᶵ⬟ࡍࡿ࡜⪃࠼ࡽࢀࡿ࠿ࢆㄝ᫂ࡋ࡞ࡉ࠸ࠋProblem 1
The “externalities” or an “external economy” refers to the impact that an agent’s actions
or behaviors has on other agents that do not go through market mechanisms. Althoughexternalities can have both positive and negative impacts, negative impacts are often
argued to be a factor leading to “market failures.”
Question 1)
Consider an industrial cluster of small-scale enterprises located in a particular region
within a developing country. Examples of industrial clusters include garment factories(the clothing industry), furniture industry, or metal parts manufacturers (themanufacturing industry). Choose one example of an industrial cluster and describe
positive and negative externalities that may exist within that cluster.
Question 2)
The “internalization” of externalities is a typical solution to problems caused byexternalities. For a negative externality you have identified in Q1, propose anarrangement or a solution mechanism that could enable its internalization. Your proposal
should include a description of the main agents of the mechanism and what actions theyneed to take.
Question 3)
The arrangement or solution mechanism that you proposed in Q2 may not function wellin developing countries. Consider main reasons why your proposal might not function
well and discuss how the existing conditions, including policies and regulations, might
be reformed to enhance the functioning of your proposal.

ၥ㢟㸰)RRG/RVV)RRG:DVWH ࡟㛵ࡍࡿ௨ୗࡢၥ࠸࡟⟅࠼ࡼࠋၥ㸯㸬㣗ᩱ࡟㛵㐃ࡋࡓၥ㢟࡜ࡋ࡚ࠊ㏵ୖᅜ࡛ࡣ୺࡟ )RRG/RVV ࡀࠊඛ㐍ᅜ࡛ࡣ୺࡟ )RRG:DVWH ࡀၥ㢟࡜࡞ࡗ࡚࠸ࡿࠋࡑࡢ⌮⏤ࢆㄝ᫂ࡋ࡞ࡉ࠸ࠋၥ㸰㸬,&7 ᢏ⾡ࡢά⏝ࡣࠊ)RRG /RVV)RRG :DVWH ࡢ๐ῶ࡟᭷ຠ࡛࠶ࡿ࡜ゝࢃࢀࡿࠋ㸯㸧)RRG /RVV ࡢ๐ῶ࡟࠾࠸࡚ ,&7 ᢏ⾡ࡀ࡝ࡢࡼ࠺࡟ά⏝࡛ࡁࡿ࠿ࠊ㸰㸧)RRG :DVWH ࡢ๐ῶ࡟࠾࠸࡚ ,&7 ᢏ⾡ࡀ࡝ࡢࡼ࠺࡟ά⏝࡛ࡁࡿ࠿ࠊලయⓗ࡞౛ࢆ࠶ࡆㄝ᫂ࡋ࡞ࡉ࠸ࠋၥ㸱㸬㏵ୖᅜ࡟࠾ࡅࡿ )RRG/RVV)RRG:DVWH ࡢ๐ῶࡣࠊ6'*V ࡢ  ┠ᶆࡢ࠺ࡕࠊ࡝ࡢࡼ࠺࡞┠ᶆ࡜㛵㐃ࡍࡿ࡜⪃࠼ࡽࢀࡿ࠿ࠊ࡛ࡁࡿࡔࡅከࡃࡢ┠ᶆ࡜ࡢ㛵ಀᛶࢆㄝ᫂ࡋ࡞ࡉ࠸ࠋゎ⟅࡟ࡣ࡝ࡢ 6'*V ࡟㛵㐃ࡍࡿࡢ࠿ࠊ)RRG/RVV ࡢ๐ῶ࡞ࡢ࠿ࠊ)RRG:DVWH ࡢ๐ῶ࡞ࡢ࠿ࢆ᫂♧ࡋ࡞ࡉ࠸ࠋᅗ  6'*V ┠ᶆ┠ᶆProblem 2Answer the following questions on Food Loss/Food Waste.
Question 1)
Among the Food related problem, Food Loss is more of a problem for developing
countries, whereas Food Waste is more of a problem for developed countries. Explainwhy.
Question 2)
ICT technologies are thought to be effective in reducing Food Loss and Food Waste.
Provide two examples:1) how a specific ICT technology can be used to reduce Food Lossand 2) how a specific ICT technology can be used to reduce Food Waste.
Question 3)
Identify and describe as many relationships as possible between Food Loss and/or FoodWaste reductions in developing countries and the 17 SDGs. In your answer, specify thenumber of SDG and whether it is Food Loss or Food Waste.



Fig.11 17 SDGsFig໰୊ 3
໰1͋Δ్্ࠃͷ೶ଜʹ͓͚ΔՈ‫ܭ‬ͷ೔ऩ z ͱҰ೔ͷ‫ۈ‬࿑࣌ؒ x ͷؔ܎Λߟ͑ΔɻՈ‫ ਺ܭ‬N ͷ೶ଜʹ͍ͭͯௐࠪ͠ɺՈ‫ ܭ‬iʢi = 1, ..., N ʣͷ೔ऩ zi ɺ‫ۈ‬࿑࣌ؒ xi ͷσʔλΛಘͨɻճ‫ࣜ࣍ͯ͠ͱࣜؼ‬ΛԾఆ͢Δɻ

z = ax + b + εz[1]͜͜Ͱ a, b ͸ఆ਺Ͱ εz ͸‫͋Ͱ߲ࠩޡ‬Δɻͳ͓ɺຊ໰ʹ͓͍ͯ͸‫ࠩޡ‬Λද֬͢཰ม਺͸͍ͣΕ΋‫͍ޓ‬ʹಠཱͰ͋Δɻ(1) ࣜ [1] ͸εz = z − ax − b[2]ͱॻ͚Δ͜ͱΛߟྀͯ͠ɺ‫ ߲ࠩޡ‬εz ͕‫ظ‬଴஋ 0 ͷਖ਼‫ن‬෼෍ΛͱΔͱ͖ɺ࠷খೋ৐๏Ͱ܎਺Λ‫ٻ‬ΊΔ͜ͱͷଥ౰ੑΛઆ໌ͤΑɻɹͳ͓ɺՈ‫ ܭ‬i Ͱͷ‫߲ࠩޡ‬ͷ஋͸ εzi ͱॻ͘ɻ·ͨɺ‫ظ‬଴஋ 0 Ͱ෼ࢄ͕ 1 ͷਖ਼‫ن‬෼෍ͷ֬཰ີ౓ؔ਺ f (ε) ͸࣍ࣜͰ༩͑ΒΕΔɻ

 2
1εf (ε) = √ exp −22π[3](2) x ͷσʔλʹ‫ظ‬଴஋ 0 ෼ࢄ 1 ͷਖ਼‫ن‬෼෍ͷ‫ ࠩޡ‬εx ͕‫·ؚ‬ΕΔ͜ͱ͕෼͔ͬͨɻͨͱ͑͹ɺՈ‫ ܭ‬i ͷ‫ۈ‬࿑࣌ؒͷਅ஋͕ɺx̄i Ͱ͋Δͱ͖ɺಘΒΕͨσʔλ xi ͸xi = x̄i + εxi[4]ͱ༩͑ΒΕΔɻ͋Δ‫؍‬ଌ஋ xi ͕ಘΒΕͨͱ͖ͷɺਅ஋ x̄i ͷ֬཰ີ౓ؔ਺ g(x̄i ) Λ‫ٻ‬ΊΑɻ(3) ࣜ [1] ʹ͓͍ͯ x Λ x̄ ʹஔ͖‫࣌ͨ͑׵‬ͷ܎਺ a, b Λ‫ٻ‬Ί͍ͨɻ͔͠͠ɺਅ஋ x̄ ͸෼͔Βͳ͍ͷͰɺ‫؍‬ଌ஋ x Λ΋͍ͪͯz = ax + b + ξ[5]ͱॻ͘ɻ‫ ߲ࠩޡ‬ξ ͸ͲͷΑ͏ͳ֬཰෼෍ʹै͏͔ɻ(4) ͜ͷͱ͖ɺa, b ͷ஋Λɺ‫؍‬ଌ஋ xi , zi (i = 1, ..., N ) Λ༻͍ͯਪఆ͢Δʹ͸ͲͷΑ͏ʹ͢Ε͹Α͍͔ɻʢ࣍ทʹଓ͘ʣ໰23 छྨͷ੡඼ X,Y,Z Λߟ͑ΔɻͦΕͧΕͷੜ࢈ྔΛ x, y, z ͱ͢Δɻ͋Δ‫ۀا‬ͷར५ p ͸ɺఆ਺ aΛ༻͍ͯɺ࣍ࣜͰ༩͑ΒΕΔɻp=z−1
22(x − a) − (y − a)4

[6]·ͨɺX,Y,Z ͷ 1 ୯Ґͷੜ࢈ʹ൐͍ɺͦΕͧΕԹࣨޮՌΨε͕ 1,1, β1 ͚ͩഉग़͞ΕΔͨΊɺ࣍ࣜͰද
͞ΕΔ੍໿Λຬͨ͞Ͷ͹ͳΒͳ͍ɻ

x+y+1
z=cβ[7]ͨͩ͠ɺc > 2a ͱ β > 0 ͸ఆ਺Ͱ͋Γɺa > 2β Ͱ͋Δͱ͢Δɻ(1) ϥάϥϯδϡͷະఆ৐਺๏Ͱར५ p Λ࠷େԽ͢Δ໰୊Λߟ͑Δɻະఆ৐਺ λ Λ༻͍ͯɺϥάϥϯδΞϯΛఆٛͤΑɻ(2) ར५ p Λ࠷େԽ͢Δ X,Y,Z ͷੜ࢈ྔ x∗ , y ∗ , z ∗ ɺ͓Αͼͦͷͱ͖ͷར५ p∗ Λ‫ٻ‬ΊΑɻ(3) ‫ۀا‬͸ X,Y Λଞ͔ࣾΒߪೖ͢ΔͨΊɺࣜ [7] Ͱද͞ΕΔ੍໿৚݅Λ
1z=cβ[8]ͱߟ͑ͯར५ p Λ࠷େԽ͢Δͱ͢Δɻ͜ͷͱ͖ͷ X,Y,Z ͷੜ࢈ྔ x∗∗ , y ∗∗ , z ∗∗ ͓Αͼར५ p∗∗Λ‫ٻ‬ΊΑɻ(4) ੍໿৚݅ࣜ [8] ͷΑ͏ʹ X ͱ Y ͷੜ࢈ʹ൐͏ԹࣨޮՌΨεഉग़੍͕໿৚݅ʹೖΒͳ͍ͱ͖ɺ͜ͷ‫( ͕ۀا‬2) Ͱ‫ٻ‬Ίͨ࠷ద஋ x∗ , y ∗ , z ∗ Λੜ࢈͢ΔΑ͏ʹ͢Δʹ͸ͲͷΑ͏ͳํࡦ͕ߟ͑ΒΕ
Δ͔ɻͳΔ΂͘۩ମతʹ 5 ߦఔ౓Ͱઆ໌ͤΑɻ(5) ͜ͷํࡦ͕‫ػ‬ೳ͢ΔͱɺԹࣨޮՌΨεͷഉग़ྔͱ‫ۀا‬ͷར५͸ͲͷΑ͏ʹมԽ͢Δ͔ɻ·ͨɺβ ͷ஋ʹΑͬͯɺԹࣨޮՌΨεΛ 1 ͚ͩ‫ݮ‬Β͢ͱ͖ͷ‫ۀا‬ͷෛ୲૿෼͸ͲͷΑ͏ʹมΘΔ͔Λߟ࡯ͤΑɻʢ໰୊ 3 ऴΘΓʣProblem 3
Question 1Consider the relationship between daily income z and working hours per day x of households ina village in a certain developing country. We conducted a survey on N households in the village,
to obtain data on the daily income zi and working hours xi for the household iʢi = 1, ..., N ʣ.
Assume the following regression equation,z = ax + b + εz[1]where a and b are constants and εz denotes an error term.
Note: For Question 1, error terms are mutually independent.
(1) Equation [1] can be written as follows.
εz = z − ax − b[2]Explain the validity of estimating the value of constants a and b by the least squares
method, when the error εz follows a normal distribution with the mean value 0. Let εzidenote the error term of the household i. The probability distribution function f (ε) of thenormal distribution with the mean value 0 and the variance of 1 is given as follows.
 2
ε1f (ε) = √ exp −22π[3](2) It was found that the data of x includes errors that follow a normal distribution with themean value 0 and the variance of 1. For example, the working hours of a household iwhose true value of working hours is x̄i , is given as follows.
xi = x̄i + εxi[4]Suppose the data xi is observed. What is the probability density function g(x̄i ) of thetrue value x̄i ?
(3) We estimate the coeﬃcients a and b for Equation [1] in which x is replaced by x̄. Since
the true value of x̄ is not known, we use the observed value x and write as follows.
z = ax + b + ξDescribe the probability density function that ξ follows.
(4) Using the observed values of xi and zi (i = 1, ..., N ), how should we estimate a and b?
(Continued on the next page.)[5]Question 2
Consider three goods X, Y and Z, whose production quantities are denoted as x, y and z,respectively. Consider a company that controls x, y and z to maximize its proﬁt p which is given
as follows.
p=z−

122(x − a) − (y − a)4[6]

where a > 0 is a constant.
Each unit production of X, Y and Z generates greenhouse gases of the amount of 1, 1 and β1 ,respectively, such that the following constraint is satisﬁed.
x+y+

1z=cβ

[7]

where c > 2a and β > 0 are constants and a > 2β.
(1) Assume the Lagrange multiplier method is used to obtain x, y and z that maximizes theproﬁt p. Using a Lagrange multiplier λ, deﬁne the Lagrangian.
(2) Derive the production values x∗ , y ∗ and z ∗ that maximizes the proﬁt, as well as the valueof proﬁt p∗ .
(3) When the company purchases X and Y from other companies, instead of Equation [7], it
optimizes the proﬁt p under the new constraint
1z = c.
β[8]Derive the production values of x∗∗ , y ∗∗ and z ∗∗ that maximizes the proﬁt under the newconstraint, as well as the value of proﬁt p∗∗ .
(4) The constraint given in Equation [8] does not consider the generation of greenhouse gasesin the production of X and Y. In about seven lines, propose and explain a strategy that
can induce this company to produce the amount x∗ , y ∗ and z ∗ as obtained in (2).
(5) How much will your proposed strategy change the amount of greenhouse gas emissions
and the proﬁt of the company? Discuss how the value of β aﬀects the cost the company
incurs in reducing greenhouse gas emissions by unit amount.
(End of Problem 3)