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15.7 a. Solve the game using backward induction starting with firm 2’s action. We saw
from Problem 15.1 part (b) that firm 2’s best-response function is
1
275 .
q
q
b. If firm 1 accommodates 2’s entry, the outcome in part (a) arises, and 1 earns
q
15.8 a. The two firms engage in Bertrand competition in homogeneous products at the
right end of the beach, leading to prices equal to marginal cost (here zero). Firm
A’s demand at the left end of the beach from Example 15.5 is
.
2 2 2 2
B A A
A
p p p
LL
qtL tL
Maximizing profit (the displayed quantity times
A
p
) yields the first-order
condition
0.
2A
p
L
tL
This implies
2
*.
2
A
tL
p
A’s profit is
3
*.
8
A
tL
b. No. B earns zero profit, so would not sink any positive investment cost to enter.
c. A’s entry-deterring strategy is not credible. From its strategy in part (a), it earns
3
*.
8
A
tL
Analytical Problems
15.9 Herfindahl index of market concentration
a. Reprising the analysis from Problem 15.2, firm i’s profit is
)( cbQbqaq iii
with associated first-order condition
2 0.
i
a b bQ c
Imposing symmetry
**
[ ( 1) ]
ii
Q n q
and solving,
*.
( 1)
i
ac
qnb
Further,
*()
,
( 1)
n a c
Qnb
*,
1
a nc
Pn
2
** ,
1
i
n a c
nbn
2
2
*,
21
n a c
CS bn
2
*( 2) .
21
n n a c
Wbn
Because firms are symmetric,
1,
i
sn
implying
2
11
.Hnnn
b. We can obtain a rough idea of the effect of merger by seeing how the variables in
part (a) change with a reduction in
.n
Per-firm output, price, industry profit, and
the Herfindahl index increase. Total output, consumer surplus, and welfare
decrease.
15.10 Inverse elasticity rule
Equation 15.2 can be rearranged as follows:
,
'/ 1,
||
i
i i i
qP
P q dP dq q
PC
P P P
,
qP
i
15.11 Competition on a circle
a. This is the indifference condition for a consumer located distance
x
from firm i:
c. Setting
*
pp
and solving for
*
p
gives the specified answer. Equilibrium price
is increasing in cost and the degree of differentiation, given by the transportation
cost and the spacing between firms (depending on their numbers).
The equilibrium is
**
12
2 3.
L
pp
Profits are
* * 2
12
(2 3) 0.444.
L
c. The best response for the high type is given in part (a) and for the low type in part
(b). The best response for firm 2 comes from maximizing its expected profit:
1 1 1
2 2 2 2 2 2
11
1 1 1 .
2 2 2 2 2
LH
p p p
p p p p p p
Note:
11
1.
2
LH
pp
p
Solving these three best responses simultaneously yields
*
141 60 0.683,
L
p
*
171 60 1.183,
H
p
*
244 60 0.733.p
d. Firm 2 earns an expected payoff of about 0.542 under complete information and
0.537 under incomplete information, and thus would prefer complete information.
e. We need to check that the low type would prefer its equilibrium profit to the
Behavioral Problem
15.13 Can competition unshroud prices?
a. Sophisticated consumers will only pay
i
s
if it is less than
,e
their personal
avoidance cost. If
e
is very small, firms will find it more profitable to try to
exploit myopic consumers instead, charging the highest value that keeps them
from voiding the purchase. They do not void if
.
ii
p s v
Hence,
.
ii
s v p
Suppose the firm posting the weakly lower price earned positive profit.
This cannot be a Nash equilibrium because its rival (which at best only makes half
the sales and indeed makes no sales if its price is strictly higher) strictly gains by
second equality in the displayed equation that posted prices can be negative (if
v
is much bigger than
c
).
separate cartridges. These prices are shrouded as they rarely show up in printer
catalog entries.
c. If firm 1 advertises, the net surplus consumers (who are all sophisticated now)
obtain from firm 2 is
*
211
( ) ( ) .
a
v e p v e c v c v c e
a
i
p
