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.
3. ަবͷఆٛ
ఆٛɿަব
࣮ݱՄೳू߹ɿ
ަবͷෆҰகɿ
ͷ ʹΑͬͯఆٛ͞ΕΔ
Ծఆ
w ۭ࣮࣍ؒݩ ͷίϯύΫτͰತͳ෦ू߹
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. ަবղ͕ຬ͖ͨͭ͢ͷެཧ
ύϨʔτ࠷దੑ
ͯ͢ͷ ʹରͯ͠ Ͱ গͳ͘ͱͭͷ ʹରͯ͠
Ͱ͋Δ ͷ ଘࡏ͠ͳ͍
ରশੑ
ަব ͕ରশͳΒ
ˎަব͕ରশͰ͋Δͱ ͳΒ
ਖ਼࣍ม͔Βͷಠཱੑ
ަব ͕ަব ͔Βޮ༻ͷਖ਼࣍ม ʹΑͬͯಘΒΕΔͱ͖
ແؔͳ݁Ռ͔Βͷಠཱੑ
̎ͭ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)