Головань С., Гуриев С., Макрушин А. Теория контрактов. Сборник задач с решениями
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t
θ θc(t) c(0) = c
(0) = 0 c
(t),c
(t) > 0
θ > 0 θ>θ θ π
(t, w) t w
0
θ
θ
t − w → max
w − θc(t) ≥ 0.
t − θc(t) → max w = θc(t)
t
FB
=(c
)
−1
(1/θ) w
FB
= θc(t
FB
)
π(t − w)+(1− π)(t − w) → max
w − θc(t) ≥ 0,
w − θc(t) ≥ 0,
w − θc(t) ≥ w − θc(t),
w − θc(t) ≥ w − θc(t).
(θ − θ)(c(t) − c(t)) ≤ 0
t ≤ t
w = θc(t),
w
= θc(t)+θc(t) − θc(t).
π(t − θc(t)) + (1 −π)(t − θc(t) − θc(t)+θc(t)) → max
t,t
.
1 − θc
(t)=0 =⇒ t = t
FB
,
π(1 −
θc
(t)) + (1 − π)(θ − θ)c
(t)=0 =⇒ t<t
FB
.
t =0 w =0 t = t
FB
w = w
FB
θ
n i
a
i
≥ 0 i =1 n
x =
a
i
i c
i
(a
i
) c
i
> 0 c
i
> 0
s
i
(x)
s
i
(x)=x
n =2 c
1
(a
1
)=a
2
1
c
2
(a
2
)=2a
2
2
(a
1
+ ···+ a
n
) − (c
1
(a
1
)+···+ c
n
(a
n
)) → max
a
1
,...,a
n
≥0
.
c
i
(a
i
)=1,i=1,...,n.
a
∗
i
=(c
i
)
−1
(1) i =1,...,n
i s
i
(x)
s
i
(a
1
+ ···+ a
n
) − c
i
(a
i
) → max
a
i
≥0
.
s
i
(x)=c
i
(a
i
)
n
i+1
s
i
(x)=
n
i+1
c
i
(a
i
).
s
i
(x)=x
c
i
(a
i
)=
s
i
(x)=1
c
i
(a
i
)=n
s
∗
i
(x) i =1,...,n
a
∗∗
i
x
∗∗
s
i
(x)
s
i
(x)=s
∗
i
(x
∗∗
)x.
x
a
1
+ a
2
− (a
2
1
+2a
2
2
) → max
a
1
,a
2
.
a
1
=1/2 a
2
=1/4 3/8=9/24
s
1
(x)=αx s
2
(x)=
(1 − α)x
α(a
1
+ a
2
) − a
2
1
→ max
a
1
a
1
= α/2
(1 − α)(a
1
+ a
2
) − 2a
2
2
→ max
a
2
a
2
=(1− α)/4
α
2
+
(1 − α)
4
−
α
2
2
− 2
(1 − α)
4
2
=
1
8
(1 + 4α − 3α
2
).
α α =2/3 7/24
N i
x
i
a
i
x
2
i
/2 a
i
≥ 0 i =1 n
x =
N
i=1
x
i
x
if
i
(x)=α
i
x + β
i
α
i
> 0
N
i=1
α
i
=1
N
i=1
β
i
=0
α
i
β
i
N
i=1
x
i
−
N
i=1
a
i
x
2
i
2
→ max
x
1
...x
n
.
1 − a
i
x
i
=0,i=1,...,N.
x
∗
i
=
1
a
i
,i=1,...,N, x
∗
=
N
i=1
1
a
i
.
W = x
∗
−
N
i=1
a
i
x
∗2
i
2
=
N
i=1
1
2a
i
.
f
i
−
a
i
x
2
i
2
→ max
x
i
.
f
i
(x)=a
i
x
i
.
f
i
(x)=α
i
x + β
i
x
∗
i
= α
i
/a
i
N
i=1
α
i
a
i
−
N
i=1
α
2
i
2a
i
→ max
α
1
...α
n
N
i=1
α
i
=1,
α
i
≥ 0,i=1,...,N.
α
i
> 0
α
i
=1−λa
i
,i=1,...,N,
N
i=1
α
i
=1 α
i
≥ 0,i=1,...,N.
λ =
N −1
N
j=1
a
j
,α
∗
i
=1−
(N −1)a
i
N
j=1
a
j
.
α
∗
i
> 0
α
i
≥ 0
K
i=1
a
i
≥ (K −1)a
K
, ,a
1
+ ...+ a
K−1
≥ (K −2)a
K
.
N
∗
=argmax
K
K : a
K
≤
a
1
+ ...+ a
K−1
K − 2
,
α
k
=
1 −
(N
∗
−1)
N
∗
i=1
a
i
a
k
, k ≤ N
∗
,
0,
k>N
∗
.
β
i
β
∗
i
β
∗
i
=0 β
i
α
i
> 0 β
i
W
SB
= x
∗
−
N
∗
i=1
a
i
x
∗2
i
2
=
N
∗
i=1
1
a
i
α
i
−
α
2
i
2
<
N
∗
i=1
1
2a
i
,
N
∗
> 1
θ
H
θ
L
θ
L
λ
θ x T
u(x, T )=θv(x) − T
v(x)=
1 − (1 − x)
2
2
.
c>0
θ
L
λ
p
i
θ
i
(F, p) F
p
θ
θv(x) − px → max
x
θv
(x)=p θ(1 −x)=p x =1− p/θ
D(p)=
1 − p
λ
θ
L
+
1−λ
θ
H
,
0 ≤ p ≤ θ
L
(1 − λ)
1 −
p
θ
H
,
θ
L
≤ p ≤ θ
H
0, p ≥ θ
H
(p − c)D(p) → max
p
(p − c)
1 − p
λ
θ
L
+
1 − λ
θ
H
→ max
p
0 ≤ p ≤ θ
L
.
D(p)
p
θ
H
θ
L
p =
1
2
λ
θ
L
+
1−λ
θ
H
+
c
2
.
(p − c)
1 −
p
θ
H
→ max
p
θ
L
≤ p ≤ θ
H
.
p =
θ
H
+c
2
θ
H
+c
2
<θ
L
θ
L
θ
H
+c
2
≤ θ
L
p =
c
2
+
1
2
λ
θ
L
+
1−λ
θ
H
c
2
+
1
2
λ
θ
L
+
1−λ
θ
H
≥ θ
L
p =
θ
H
+c
2
c
2
+
1
2
λ
θ
L
+
1−λ
θ
H
<θ
L
<
θ
H
+c
2
p
1
=
θ
H
+c
2
p
2
=
c
2
+
1
2
λ
θ
L
+
1−λ
θ
H
p = p
1
=⇒ π
1
=(p
1
− c)(1 − λ)
1 −
p
1
θ
H
=
1 − λ
4θ
H
(θ
H
− c)
2
,
p = p
2
=⇒ π
2
=(p
2
− c)
1 − p
2
λ
θ
L
+
1 − λ
θ
H
=
λ
θ
L
+
1 − λ
θ
H
1
4
1
λ
θ
L
+
1−λ
θ
H
− c
2
.
λ
c
2
+
1
2
λ
θ
L
+
1−λ
θ
H
θ
L
π
1
→ 0
π
2
→
1
4θ
L
(θ
L
− c)
2
> 0 λ
D
L
=
1 −
p
θ
L
, 0 ≤ p ≤ θ
L
0, p>θ
L
D
H
=
1 −
p
θ
H
, 0 ≤ p ≤ θ
H
0, p>θ
H
p
L
=
θ
L
+c
2
p
H
=
θ
H
+c
2
π =
λ
4θ
L
(θ
L
− c)
2
+
1−λ
4θ
H
(θ
H
− c)
2
F +(p − c)(λD
L
(p)+(1−λ)D
H
(p)) → max
F,p
θ
L
v(D
L
(p)) − (F + pD
L
(p)) ≥ 0,
θ
H
v(D
H
(p)) − (F + pD
H
(p)) ≥ 0.
F ≤
1
2θ
i
(p − θ
i
)
2
i ∈{L, H}
1
2θ
(p − θ)
2
θ
=
1
2
1 −
p
2
θ
2
> 0
F ≤
1
2θ
L
(p − θ
L
)
2
1
2θ
L
(p − θ
L
)
2
+(p − c)
1 − p
λ
θ
L
+
1 − λ
θ
H
→ max
p
1
2θ
L
2(p − θ
L
)+1− (2p − c)
λ
θ
L
+
1 − λ
θ
H
=0,
p
2λ − 1
θ
L
+
2 − 2λ
θ
H
= c
λ
θ
L
+
1 − λ
θ
H
.
p =
λθ
H
+(1− λ)θ
L
(2λ − 1)θ
H
+(2− 2λ)θ
L
c.
p<θ
L
F =
1
2θ
L
(p − θ
L
)
2
θ
L
p
∗
= c, F
∗
=
1
2θ
H
(θ
H
− c)
2
.
F
∗
p ≥ θ
L
p ≥ θ
L
θ
L
λ
(x
L
,T
L
) (x
H
,T
H
)
λ(T
L
− cx
L
)+(1− λ)(T
H
− cx
H
) → max
x
L
,x
H
,T
L
,T
H
,
θ
L
v(x
L
) − T
L
≥ 0,
θ
H
v(x
H
) − T
H
≥ 0,
θ
L
v(x
L
) − T
L
≥ θ
L
v(x
H
) − T
H
,
θ
H
v(x
H
) − T
H
≥ θ
H
v(x
L
) − T
L
.
θ
H
v(x
H
)−T
H
≥ 0
θ
H
v(x
H
) − T
H
≥ θ
H
v(x
L
) − T
L
θ
L
v(x
L
) − T
L
≥ θ
L
v(x
H
) − T
H
λ(T
L
− cx
L
)+(1−λ)(T
H
− cx
H
) → max
x
L
,x
H
,T
L
,T
H
,
θ
L
v(x
L
) − T
L
≥ 0,
θ
H
v(x
H
) − T
H
≥ θ
H
v(x
L
) − T
L
,
x
H
≥ x
L
.
x
H
=1−
c
θ
H
,
x
L
=1−
λc
θ
L
− (1 − λ)θ
H
=1−
λc
λθ
L
− (1 − λ)(θ
H
− θ
L
)
<x
∗
L
=1−
c
θ
L
,
x
∗
L
θ
H
p>c
θ
L
P
W
U
T
= const
U
B
= const
π = const
P W
v − θ
B
P − W,
v − θ
T
P − W,
0 <θ
B
<θ
T
W
λ
c
(P, W)
λ θ
B
θ
T
c
(P, W)
v − θ
i
P − W =const
W
W P