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U10. (a) πL = M – 0.1Q
πH = M – 0.16Q
(d) Oceania’s expected net benefit is
1 1 2 2
0.6(2 ) 0.4(2 ).B Q M Q M= - + -
(e) Assuming IC1 binds:
Assume PC2 binds:
Substituting the PC2 equation into the IC1 equation, we have
So our minimum required payments are, in terms of Q1 and Q2:
(f) First, consider PC1:
Substituting for M1 from (e), we have
The quantity Q2 is not negative, so PC1 is automatically satisfied.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Next, consider IC2:
Substituting for M1 and M2 using the results derived in part (e), we have
Thus, as long as the final contracts specify that the low-cost firm should produce a (weakly) higher
quantity than the high-cost firm, which we should expect, then the constraint IC2 will be automatically
satisfied.
(g) Substituting our results from part (e) into our results from part (d):
1 1 2 2
1 1 2 2 2
1 1 2 2 2
1 1 2 2
0.6(2 ) 0.4(2 )
0.6(2 [0.1 0.06 ]) 0.4(2 0.16 )
1.2 0.06 0.036 0.8 0.064
1.2 0.06 0.8 0.1
B Q M Q M
Q Q Q Q Q
Q Q Q Q Q
Q Q Q Q
= - + -
= - + + -
= - - + -
= - + -
(h) This function must be maximized with respect to both Q1 and Q2. The first-order
condition for Q1 is
1/2
1
1
1/2
1
1
1
1
1.2(1/ 2) 0.06 0
0.06 0.6
1
0.1
10
100
dB Q
dQ
Q
Q
Q
Q
-
-
= - =
Þ =
Þ =
Þ =
Þ =
The first-order condition for Q2 is
1/2
2
2
1/2
2
2
2
2
0.8(1/ 2) 0.1
0.1 0.4
1
0.25
4
16
dB Q
dQ
Q
Q
Q
Q
-
-
= -
Þ =
Þ =
Þ =
Þ =
So the optimal menu of contracts will specify Q1 = 100 and Q2 = 16. Note that this satisfies the assumption
we made in part (f), that Q1 ≥ Q2.
(j) The expected net benefit is, from part (d):
1 1 2 2
0.6(2 ) 0.4(2 )
0.6(2 100 10.96) 0.4(2 16 2.56)
7.6
B Q M Q M= - + -
= - + -
=
(k) As in Exercise S11, the low-cost contract includes extra profit to induce the low-cost firm
While the carrot is smaller as p increases, the stick to discourage a low-cost BMA from taking the
(b) IC1: M1 – 0.2Q1 ≥ M2 – 0.2Q2
(d) Oceania’s expected net benefit is
1 1 2 2
0.4(2 ) 0.6(2 ).B Q M Q M= - + -
(e) Assuming IC1 binds
Assume PC2 binds
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Substituting the PC2 equation into the IC1 equation, we have
So our minimum required payments are, in terms of Q1 and Q2:
(f) First, consider PC1:
Substituting for M1 from part (e), we have
The quantity Q2 is not negative, so PC1 is automatically satisfied.
Next, consider IC2:
Substituting for M1 and M2 using the results derived in part (e), we have
Thus, as long as the final contracts specify that the low-cost firm should produce a (weakly) higher
quantity than the high-cost firm, which we should expect, then the constraint IC2 will be automatically
satisfied.
(g) Substituting our results from part (e) into our results from part (d):
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
1 1 2 2
1 1 2 2 2
1 1 2 2 2
1 1 2 2
0.4(2 ) 0.6(2 )
0.4(2 [0.2 0.18 ]) 0.6(2 0.38 )
0.8 0.08 0.072 1.2 0.228
0.8 0.08 1.2 0.3
B Q M Q M
Q Q Q Q Q
Q Q Q Q Q
Q Q Q Q
= - + -
= - + + -
= - - + -
= - + -
(h) The first-order condition for Q1 is
1/2
1
1
1/2
1
1
1
1
0.8(1/ 2) 0.08 0
0.08 0.4
1
0.2
5
25
dB Q
dQ
Q
Q
Q
Q
-
-
= - =
Þ =
Þ =
Þ =
Þ =
The first-order condition for Q2 is
1/2
2
2
1/2
2
2
2
2
1.2(1/ 2) 0.3
0.3 0.6
1
0.5
2
4
dB Q
dQ
Q
Q
Q
Q
-
-
= -
Þ =
Þ =
Þ =
Þ =
So the optimal menu of contracts will specify Q1 = 25 and Q2 = 4. Note that this satisfies the assumption
that Q1 ≥ Q2.
(j) The expected net benefit is, from part (d):
1 1 2 2
0.4(2 ) 0.6(2 )
0.4(2 25 5.72) 0.6(2 4 1.52)
3.2
B Q M Q M= - + -
= - + -
=
(k) Comparing parts (f) and (g) from Exercise S11 and this exercise, we see that when the
per-unit costs increase for each type of BMA, the optimal amount that Oceania will order in each contract
(b) The participation constraints are
1 1 1 1
2 2 2 2
3 3 3 3
: Q 0
: Q 0
: Q 0
P M c
P M c
P M c
- ³
- ³
- ³
(c) The incentive-compatibility constraints are
12 1 1 1 2 1 2
13 1 1 1 3 1 3
21 2 2 2 1 2 1
23 2
: for type 1 not to mimic type 2
: for type 1 not to mimic type 3
: for type 2 not to mimic type 1
:
IC M c Q M c Q
IC M c Q M c Q
IC M c Q M c Q
IC M c
- ³ -
- ³ -
- ³ -
-2 2 3 2 3
31 3 3 3 1 3 1
32 3 3 3 2 3 2
for type 2 not to mimic type 3
: for type 3 not to mimic type 1
: for type 3 not to mimic type 2
Q M c Q
IC M c Q M c Q
IC M c Q M c Q
³ -
- ³ -
- ³ -
(d) Oceania’s net benefit, B, is
1 1 1 2 2 2 3 3 3
(2 ) (2 ) (2 ). (1)B p Q M p Q M p Q M= - + - + -
(e) By P3, we have
3 3 3. (2)M c Q³
Next, combining P3 and IC23:
2 2 2 2 3 3 3 (3)M c Q c Q c Q³ - +
Finally, combine P3, IC23, and IC12 to obtain
1 1 1 1 2 2 2 2 3 3 3. (4)M c Q c Q c Q c Q c Q³ - + - +
(f) From the objective function (1), we see that to maximize expected benefit, for whatever
(g) With equalities in the three selected constraints, to verify that the others are fulfilled,
consider:
IC12: From (4) and (2) holding as equations, we have
1 3 1 1 1 2 2 2 2 3
1 1 1 3 1 3 1 3 1 1 1 3
1 1 1 2 2 2 2 3 1 1 1 3
( ) ( )
M M c Q c Q c Q c Q
M c Q M c Q M M c Q c Q
c Q c Q c Q c Q c Q c Q
- = - + -
Þ - - - = - - +
= - + - - +
2 1 2 3
2 1 2 3
( )( ) 0
since and we are assuming
c c Q Q
c c Q Q
= - - >
> >
IC32: From (3) and (2) holding as equations, we have
3 2 3 3 2 2 2 3 3 3
3 3 3 2 3 2 3 2 3 3 3 2
3 3 2 2 2 3 3 3 3 3 3 2
( ) ( )
M M c Q c Q c Q c Q
M c Q M c Q M M c Q c Q
c Q c Q c Q c Q c Q c Q
- = - + -
Þ - - - = - - +
= - + - - +
3 2 2 3
3 2 2 3
( )( ) 0
since and we are assuming
c c Q Q
c c Q Q
= - - >
> >
IC21: From (4) and (3) holding as equations, we have
2 1 1 1 1 2
2 2 2 1 2 1 2 1 2 2 2 1
1 1 1 2 2 2 2 1
2 1 1 2
( ) ( )
( )( ) 0
M M c Q c Q
M c Q M c Q M M c Q c Q
c Q c Q c Q c Q
c c Q Q
- =- +
Þ - - - = - - +
=- + - +
= - - >
2 1 1 2
since and we are assuming c c Q Q> >
IC31: We already have an expression for M1 − M3, so
3 1 1 1 1 2 2 2 2 3
3 3 3 1 3 1 3 1 3 3 3 1
1 1 1 2 2 2 2 3 3 3 3 1
( ) ( )
M M c Q c Q c Q c Q
M c Q M c Q M M c Q c Q
c Q c Q c Q c Q c Q c Q
- =- + - +
Þ - - - = - - +
=- + - + - +
3 1 3 1 1 2 2 2 3
3 1 2 3 2 3 1 1 2 2 2 3
3 1 1 2 3 2
( ) ( ) ( )
( ) ( ) ( ) ( )
( )( ) ( )
c Q Q c Q Q c Q Q
c Q Q c Q Q c Q Q c Q Q
c c Q Q c c
= - - - - -
= - + - - - - -
= - - + - 2 3
3 2 1 1 2 3
( ) 0
since and we are assuming
Q Q
c c c Q Q Q
- >
> > > >
P2: Since (3) holds as an equation, we have
2 2 2 3 2 3
( ) 0. (5)M c Q c c Q- = - >
P1: Similarly, since (4) holds as an equation, we have
1 1 1 2 1 2 3 2 3
( ) ( ) 0. (6)M c Q c c Q c c Q- = - + - >
(h) Substituting the M1, M2, and M3 when (4), (3), and (2) hold as equalities into the objective
function (1):
1 1 1 1 1 2 2 2 2 3 3 3 2 2 2 2 2 3 3 3 3 3 3 3
1 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 3 2 2 2 2 3 3 3 3 3 3
[2 ( )] [2 ( )] [2 ( )]
2( ) ( ) ( ) ( )
B p Q c Q c Q c Q c Q c Q p Q c Q c Q c Q p Q c Q
p Q p Q p Q p c Q c Q c Q c Q c Q p c Q c Q c Q p c Q
= - - + - + + - - + + -
= + + - - + - + - - + -
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(i) To maximize B, find the first-order conditions for each Qi:
1 1
11
2 2 1 2 1
22
3 3 1 2 3 2
33
10
1( ) 0
1( )( ) 0
Bp c
QQ
Bp c p c c
QQ
Bp c p p c c
QQ
é ù
¶= - =
ê ú
¶ê ú
ë û
é ù
¶= - - - =
ê ú
¶ê ú
ë û
é ù
¶= - - + - =
ê ú
¶ê ú
ë û
These first-order conditions imply that the optimal values of each Qi are
2
1 1
2
1
2 2 2 1
2
2
1 2
3 3 3 2
3
( ) (7)
( ) (8)
( ) (9)
Q c
p
Q c c c
p
p p
Q c c c
p
-
-
-
=
é ù
= + -
ê ú
ë û
é ù
+
= + -
ê ú
ë û
(j) From (8) and (7) we see that
2 2
1 1 2 2
( ) ( ) .Q c c Q
- -
= > >
That is, it will always be true that Q1 > Q2. From (9) and (8) we see that for Q2 > Q3 to hold it must be true
that
1 2 1
3 3 2 2 2 1
3 2
1 2 1
3 2 2 1
3 2
1
3 2 2 1
3 2
3 2 1 3
2 1 2
( ) ( )
1 ( ) ( )
1
( ) ( )
( )
( )
p p p
c c c c c c
p p
p p p
c c c c
p p
p
c c c c
p p
c c p p
c c p
+
+ - > + -
é ù
+
Þ + - > -
ê ú
ë û
Þ - > -
-
Þ >
-
