This document describes a maximization problem involving two functions uA and uB defined on the unit square. It finds that the maximum value occurs when uA and uB are both 1/2, which corresponds to the point (1/2, 1/2) in their domain. The document provides the definitions of uA and uB, defines the constraint set, derives the optimal solution, and verifies that it satisfies the necessary conditions for an extremum.

1. ήʔϜཧ࿦#4*$ԋश
ަ‫͚͓ʹࡁܦ׵‬Δަবղ

2. φογϡަবղ
໰୊
ղ౴

3. ަব໰୊ͷఆٛ
ఆٛɿަব໰୊͸
࣮‫ݱ‬Մೳू߹ɿ
ަবͷෆҰக఺ɿ
ͷ૊ ʹΑͬͯఆٛ͞ΕΔ
Ծఆ
w ͸࣍‫ۭؒ਺࣮ݩ‬ ͷίϯύΫτͰತͳ෦෼ू߹
w
w ͷ఺ ͕ଘࡏͯ͠
U
d = (d1
, d2
)
(U, d)
U ℜ2
d ∈ U
U u = (u1
, u2
) ui
di
(i = 1, 2)
0
u2
u1
(d1
, d2
) = (20, 20)
U
100
100

4. ަবղ͕ຬͨ͢΂͖ͭͷެཧ
ύϨʔτ࠷దੑ
͢΂ͯͷ ʹରͯ͠ Ͱ গͳ͘ͱ΋ͭͷ ʹରͯ͠
Ͱ͋Δ ͷ఺ ͸ଘࡏ͠ͳ͍
ରশੑ
ަব໰୊ ͕ରশͳΒ͹
ˎަব໰୊͕ରশͰ͋Δͱ͸ ͳΒ͹
ਖ਼࣍ม‫͔׵‬Βͷಠཱੑ
ަব໰୊ ͕ަব໰୊ ͔Βޮ༻ͷਖ਼࣍ม‫׵‬ ʹΑͬͯಘΒΕΔͱ͖
ແؔ܎ͳ݁Ռ͔Βͷಠཱੑ
̎ͭͷަব໰୊ ͱ ʹ͓͍ͯ ͱ͢Δ
΋͠ ͳΒ͹ Ͱ͋Δ
i = 1, 2 ti
≥ fi
(U, d) i
ti
fi
(U, d) U t = (t1
, t2
)
(U, d) f1
(U, d) = f2
(U, d)
d1
= d2
(u1
, u2
) ∈ U (u2
, u1
) ∈ U
(U′

, d′

) (U, d) ui′

= αi
ui
+ βi
, αi
0 (i = 1,2)
fi
(U′

, d′

) = αi
fi
(U, d) + βi
, αi
0 (i = 1,2)
(U, d) (T, d) T ⊂ U
f(U, d) ∈ T f(T, d) = f(U, d)

5. ަবղ͕ຬͨ͢΂͖ͭͷެཧ
ఆཧ
ަব໰୊ ͷ଒ ͱ͢Δ
ͭͷެཧΛຬͨ͢ަব໰୊ͷղ ͕Ұҙʹଘࡏ͠
͢΂ͯͷަব໰୊ ʹରͯ͠ ͸
ͷղͰ͋Δ
͜ͷղΛφογϡަবղͱ͍͏
·ͨ Λφογϡੵͱ͍͏
(U, d) B
f : B → ℜ2
(U, d) f(U, d)
max
u∈U:u≥d
(u1
− d1
)(u2
− d2
)
(u1
− d1
)(u2
− d2
)

6. ̎ͭͷ෼ׂՄೳͳࡒ ࡒ ࡒ Λަ‫͢׵‬Δࢢ৔Λߟ͑Δ
͜͜Ͱ͸ ਓͷফඅऀ ফඅऀ ফඅऀ# ͕͓Γ ॳ‫ظ‬อ༗ΛͦΕͧΕ
ิ଍ɿ ͱॻ͍ͨͱ͖ ফඅऀ ͷॳ‫ظ‬อ༗ͷ͜ͱͰ͋Γ ͕ࡒͷྔ ͕ࡒͷྔʹͳΔ
֤ফඅऀͷޮ༻ؔ਺Λ ͱ͢Δ
͜ͷਓͷফඅऀ͕ަব͢Δͱߟ͑ φογϡަবղʹΑΔࡒͷ෼഑Λ‫ٻ‬ΊΑ
ͳ͓ ަবܾ྾࣌͸ॳ‫ظ‬อ༗Λ࣋ͬͨ··ͱ͢Δ
eA
= (eA
1 , eA
2 ) = (1,0) eB
= (eB
1 , eB
2 ) = (0,1)
ei
= (ei
1, ei
2) i ei
1 ei
2
uA
(x1, x2) = x
1
2
1
x
1
2
2
uB
(y1, y2) = y
1
2
1
y
1
2
2
໰୊

7. ̎ͭͷ෼ׂՄೳͳࡒ ࡒ ࡒ Λަ‫͢׵‬Δࢢ৔Λߟ͑Δ
͜͜Ͱ͸ ਓͷফඅऀ ফඅऀ ফඅऀ# ͕͓Γ ॳ‫ظ‬อ༗ΛͦΕͧΕ
ิ଍ɿ ͱॻ͍ͨͱ͖ ফඅऀ ͷॳ‫ظ‬อ༗ͷ͜ͱͰ͋Γ ͕ࡒͷྔ ͕ࡒͷྔʹͳΔ
֤ফඅऀͷޮ༻ؔ਺Λ ͱ͢Δ
͜ͷਓͷফඅऀ͕ަব͢Δͱߟ͑ φογϡަবղʹΑΔࡒͷ෼഑Λ‫ٻ‬ΊΑ
ͳ͓ ަবܾ྾࣌͸ॳ‫ظ‬อ༗Λ࣋ͬͨ··ͱ͢Δ
ަবܾ྾࣌ͷޮ༻͸ҎԼʹͳΔɿ
eA
= (eA
1 , eA
2 ) = (1,0) eB
= (eB
1 , eB
2 ) = (0,1)
ei
= (ei
1, ei
2) i ei
1 ei
2
uA
(x1, x2) = x
1
2
1
x
1
2
2
uB
(y1, y2) = y
1
2
1
y
1
2
2
dA
= uA
(1,0) = 1
1
20
1
2 = 0 dB
= uB
(0,1) = 0 ⋅ 1 = 0
ղ౴

8. ֤ফඅऀͷޮ༻ؔ਺Λ ͱ͢Δ
࣮‫ݱ‬Մೳू߹Λߟ͑ΔύϨʔτ࠷దͷ৚݅ΑΓ ֤ফඅऀͷ‫ݶ‬ք୅ସ཰͕౳͘͠ͳΔ͜ͱ͔Β
·ͨ ॳ‫ظ‬อ༗Λաෆ଍ͳ͘ফඅ͢Δ͜ͱ͕࠷దͳͨΊ ҎԼͷࣜΛຬͨ͢
Ώ͑ʹ ͜Ε͸ ύϨʔτ࠷దͳ഑෼Ͱ͸ ࡒͱࡒͷফඅྔ͕Ұக͢Δ͜ͱΛҙຯ
·ͨ ΋੒Γཱͭ
ͱ͓͚͹ ΑΓ
Ͱ͋Δ͜ͱʹ஫ҙͯ͠
uA
(x1, x2) = x
1
2
1
x
1
2
2
uB
(y1, y2) = y
1
2
1
y
1
2
2
∂uA
/∂x1
∂uA/∂x2
=
∂uB
/∂y1
∂uB/∂y2
⇔
x1
x2
=
y1
y2
x1 + y1 = 1 x2 + y2 = 1
x1
x2
=
1 − x1
1 − x2
⇔ x1 = x2
y1 = y2
x1 = x2 = t uA
= x
1
2
1
x
1
2
2
= t
1
2t
1
2 = t uB
= y
1
2
1
y
1
2
2
= (1 − x1)
1
2(1 − x2)
1
2 = (1 − x1) = (1 − t) uB
= 1 − uA
0 ≤ t ≤ 1 0 ≤ uA
≤ 1 0 ≤ uB
≤ 1
ղ౴

9. ͸ύϨʔτ࠷దͷ෦෼Ͱ͋Γ ࡒͷഁ‫غ‬Λߟྀͯ͠ ࣮‫ݱ‬Մೳू߹͸
ަবͷෆҰக఺
φογϡੵ࠷େԽ໰୊Λߟ͑Δ
TU
ύϨʔτ࠷దੑ͔Β Λຬͨ͢ͷͰ
͜Ε͸্ʹತͳೋ࣍ؔ਺ͷ࠷େԽ໰୊ͳͷͰ ࠷దղ͸ ͜ͷͱ͖ ʹ஫ҙͯ͠
͔ͩΒ φογϡަবղͰಘΒΕΔ഑෼͸
uB
= 1 − uA
U = {(uA
, uB
)|uB
≤ 1 − uA
, 0 ≤ uA
, 0 ≤ uB
}
(dA
, dB
) = (0, 0)
max
uA
,uB
(uA
− dA
)(uB
− dB
) = max
uA
,uB
uA
uB
uB
≤ 1 − uA
, 0 ≤ uA
, 0 ≤ uB
uB
= 1 − uA
max
uA
uA
(1 − uA
)
uA
=
1
2
uB
=
1
2
x1 = x2
uA
(x1, x2) = x
1
2
1
x
1
2
2
= x1 =
1
2
(x1, x2) =
(
1
2
,
1
2)
, (y1, y2) =
(
1
2
,
1
2)
ղ౴
0
uB
uA
1
1

10. ิ଍ ‫ݶ‬ք୅ସ཰.34
0A
2
1
ফඅऀͷແࠩผ‫ۂ‬ઢ
ޮ༻ਫ४͕ಉ͡ࡒͷ૊
0A
2
1
ࡒΛ୯Ґ૿΍ͨ͠ͱ͖ʹ
ޮ༻ਫ४ΛอͭͨΊʹ
‫ݮ‬Β͢ࡒͷྔʹ୅ସ཰
1
୅ସ཰
0A
2
1
઀ઓͷ܏͖ʹ‫ݶ‬ք୅ସ཰
‫ݶ‬ք୅ସ཰ແࠩผ‫ۂ‬ઢͷ܏͖
−
dx2
dx1
=
∂u/∂x1
∂u/∂x2
dx1
dx2

11. .34ʹࡒͷ‫ݶ‬քޮ༻ࡒͷ‫ݶ‬քޮ༻
ͷಋग़
‫ݶ‬քޮ༻ͱ͸ ୯ҐͷࡒΛ૿΍ͨ͠ͱ͖ͷޮ༻ͷ૿ՃྔͰ͋Δ
૯ޮ༻ͷ૿ՃྔΛ ͱ͓͘ͱ ͚֤ͩࡒ͕มԽͨ͠ͱ͢Δͱ
ࠓ ޮ༻ਫ४ΛมԽͤ͞ͳ͍ͱ͖ ແࠩผ‫ۂ‬ઢ্ͷͱ͖ Ͱ͋Δ͔Β
−
dx2
dx1
=
∂u/∂x1
∂u/∂x2
du dx1 dx2
du =
∂u
∂x1
dx1 +
∂u
∂x2
dx2
du = 0
0 =
∂u
∂x1
dx1 +
∂u
∂x2
dx2 ⇔ −
dx2
dx1
=
∂u/∂x1
∂u/∂x2
ิ଍ ‫ݶ‬ք୅ସ཰.34

12. ิ଍ ΤοδϫʔεɾϘοΫε
0A
2
1
0B
2
1
ফඅऀͷແࠩผ‫ۂ‬ઢ
ޮ༻ਫ४͕ಉ͡ࡒͷ૊
0A
2
1
0B
2
1
ফඅऀ#ͷແࠩผ‫ۂ‬ઢ
ޮ༻ਫ४͕ಉ͡ࡒͷ૊
0A
2
1
0B
2
1
ফඅऀ#ͷޮ༻ਫ४Λ
Լ͛Δ͜ͱͳ͘
ফඅऀͷޮ༻ਫ४Λߴ͘Ͱ͖Δ
ύϨʔτվળͱ͍͏
ӈ্΄Ͳޮ༻͸ߴ͍
ࠨԼ΄Ͳޮ༻͸ߴ͍
ύϨʔτվળͰ͖ͳ͍ঢ়ଶΛ
ύϨʔτޮ཰ͱ͍͍
ແࠩผ‫ۂ‬ઢ͸઀͢Δ‫ݶ‬ք୅ସ཰͕Ұக

13. ήʔϜཧ࿦#4*$ԋश
ަ‫͚͓ʹࡁܦ׵‬Δަবղ