978-0393919684 Chapter 13 Solution Manual Part 5

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.

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

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

= – + –

= – + + –

= – – + –

= – + –

(h) This function must be maximized with respect to both Q1 and Q2. The first-order

condition for Q1 is

1/2

1

1/2

1

1.2(1/ 2) 0.06 0

0.06 0.6

1

0.1

10

100

dB Q

dQ

Q

–

= – =

Þ =

The first-order condition for Q2 is

1/2

2

1/2

2

0.8(1/ 2) 0.1

0.1 0.4

1

0.25

4

16

dB Q

dQ

Q

–

= –

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

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):

1 1 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

= – + –

= – + + –

= – – + –

= – + –

(h) The first-order condition for Q1 is

1/2

1

1/2

1

0.8(1/ 2) 0.08 0

0.08 0.4

1

0.2

5

25

dB Q

dQ

Q

–

= – =

Þ =

The first-order condition for Q2 is

1/2

2

1/2

2

1.2(1/ 2) 0.3

0.3 0.6

1

0.5

2

4

dB Q

dQ

Q

–

= –

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

P M c

– 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

– 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

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

( )( ) 0

since and we are assuming

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

( )( ) 0

since and we are assuming

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

= – – + – + + – – + + –

= + + – – + – + – – + –

(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

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 >

–