18
Bargaining Problems
1.
(a)
v
∗
=50
,
000;
u
∗
J
=
u
∗
R
=25
,
000;
t
=15
,
000.
(b) Solving max
x
60
,
000
−
x
2
+ 800
x
yields
x
∗
=400
. Th
isimp
l
ies
v
∗
=
220
,
000,
u
∗
J
=
u
∗
R
=110
,
000,
v
J
=
−
100
,
000, and
v
R
= 320
,
000. Thus,
t
= 210
,
000.
(c) From above,
x
∗
= 400 and
v
∗
=220
,
000.
u
∗
J
=40
,
000 + (220
,
000
−
40
,
000
−
20
,
000)
/
4=80
,
000 and
u
∗
R
=20
,
000+(3
/
4)(220
,
000
−
60
,
000) =
140
,
000. This implies
t
= 180
,
000.
2.
(a) The surplus with John working as a programmer is 90
,
000
−
w
.Th
e
surplus with him working as a manager is
x
−
40
,
000
−
w>
110
,
000
−
w
.
Thus, the maximal joint value is attained by John working as a manager.
John’s overall payo
f
is
w
+
π
J
[
x
−
40
,
000] which is equal to (1
−
π
J
)
w
+
π
J
[
x
−
40
,
000]. The
f
rm’s payo
f
is
π
F
[
x
−
40
,
000
−
w
]. Knowing that
John’s payo
f
must equal
t
−
40
,
000, we
f
nd that
t
=[1
−
π
J
][
w
−
40
,
000]+
π
j
x
.
(b) John should undertake the activity that has the most impact on
t
,and
hence his overall payo
f
, per time/cost. A one-unit increase in
x
will raise
t
by
π
J
. A one unit increase in
w
raises
t
by 1
−
π
J
.A
s
sum
ingtha
t
x
and
w
can be increased at the same cost, John should increase
x
if
π
j
>
1
/
2;
otherwise, he should increase
w
.
3.
(a)
x
,t
=0
,
and
u
1
=
u
2
.
u
1
u
2