174 M
. Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
ˆ ˆ
E Fux , Sx . E
Fux , Sx 26
f g
f g
Sx 1
1 Sx
2 2
1 2
ˆ ˆ
for any strictly convex function Fu , i.e. for any strictly convex function Fu ,
ˆ ˆ
E
Fux , sfx , s ds .
E
Fux , sfx , s ds
1 1
2 2
where f x, s is the density function of Sx evaluated at s.
Proof. See Appendix A.
We may then state the following result. ˆ
¯
Proposition 2. Let Q and Q be the total first period output and the total myopic output
1 1
]
]
respectively . Then, the signal S 5 S
is more informative than the signal S 5 S .
ˆ Q
Q
1 1
] ]
ˆ ˆ ˆ
ˆ
]
Proof. By lemma 3 it suffices to prove that E FuQ , S . E FuQ , S for any
f g
f g
ˆ S
1 S
1
strictly convex function F. ]
] ˆ
ˆ
]
Consider the signal S 5 S and S 5 S . By proposition 1, Q .Q
and since by
ˆ Q
Q 1
1
1 1
ˆ Proposition 2, E Fux, Sx
is an increasing function of x, whenever F is strictly
f g
ˆ S
] ]
ˆ ˆ ˆ
ˆ
]
convex, then E FuQ , S . E FuQ , S for any strictly convex function F. j
f g
f g
ˆ S
1 S
1
4. Experimentation in monopoly versus duopoly
We compare next the monopoly and the duopoly first period experimental behavior.
i i
i
ˆ ¯
ˆ We stated in Proposition 1 that a firm experiments whenever q q , where q is the
1 1
1 i
¯ optimal output of the two period expected profits and q is the myopic choice. We will
1 i
i j
j
ˆ ¯
ˆ ¯
say that firm i experiments at least as much as firm j if q 2 q q 2 q . Our main
1 1
1 1
findings in this section are summarized in the following Proposition.
Proposition 3. The monopoly will experiment more than the duopoly whenever the
˜ demand
’s unknown parameter u is not too ‘diffuse’, i.e. whenever its precision h is big
9
enough .
˜ The intuition of this result is clear. If u is quite precise so is the market signal, so that
by increasing output with respect to the myopic choice, each firm makes the market ˜
price even a more informative signal and hence posterior beliefs u are more accurate. But, since the price is commonly observed, each firm is also affecting its rival updating.
Therefore, firms face a strategic informational choice: how informative to make the publicly observed market signal. This choice will depend on the informational properties
of market competition. In other words, firms’ experimental behavior will be based on
9
ˆ Notice that in this case both, the market signal and the ex-ante posterior beliefs u, are also quite precise. In
2 2
˜ particular, by Eq. 25 the variance of the market signal a 2P Q is tQ 1h htQ , and by Eq. 7
1 1
1 2
2 4
˜ ˆ
ˆ the variance of u is t Q h var a 2P Q , so that both variances decrease in h.
1 1
M . Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
175
how information and competition in the market are related. Since under Cournot competition, goods are substitutes and since information or more informative signals
10
increases each firm’s expected future production, they become informationally
substitutes, so that a better informed firm is a more unpleasant competitor. However if the uncertainty is large, market signals will not be very informative and
posteriors will not be too precise at the myopic output choice. Under this situation the above strategic informational choice although operating may be less important than the
ˆ own learning incentive. In fact, it could be the case that E[VuQ ] be convex in Q,
meaning that expected second-period profits increase more than proportionally to the increase in aggregated output. Hence, even though each firm experiments less than the
monopoly the combined experimentation of both firms may exceed that of the monopoly.
In sharp contrast, when the market signal is quite precise even for the myopic output choice, firms in a duopoly are not so willing to experiment, since by doing so they make
ˆ the posterior u very precise but for both themselves and for their rivals. Obviously, in
this situation, a monopoly will acquire a lot of information by experimenting. Therefore, in this case, information acquisition under Cournot is smaller than under monopoly, i.e.
the experimental behavior of firms is softer under quantity–duopoly competition than under monopoly. Thus, we have:
˜
Corollary. If the random demand ’s parameter u is not too ‘diffuse’ whenever the
monopoly does not experiment neither does it the duopoly ; but if the duopoly does not
experiment the monopoly may do it .
The proof of Proposition 3 is in three steps. First, Lemma 4 displays the experimental monopoly optimal choice. Second, Proposition 4 shows that the monopoly will
ˆ experiment at least as much as the duopoly whenever E[VuQ ] is concave in Q.
Finally, Lemma 5 establishes that a sufficient condition for this to happen is that the ˜
random slope parameter u be sufficiently precise. Thus, consider first the same model as the one described above, but for a monopolist
M M
ˆ ˆ
firm. Let Q and Q
be the monopoly solution for the first period and the myopic
1 1
M
ˆ output respectively. Denote by V u the monopoly expected profits for the second
M 2
M
ˆ ˆ
ˆ period. It is easily verified that V u 5a 4u and hence V u is strictly convex.
M
ˆ For any first period monopoly output x, let Sx 5a 2P x, and let E [V ux, Sx] be
1 S
M
ˆ the expectation of V u under the choice of x. By the proof of lemma 3,
M
ˆ E [V ux, Sx] is increasing in x. In Appendix A it is shown:
S
x
M M
ˆ ¯
Lemma 4. Q . Q 5a 2m.
1 1
i 1
2
Next, recall that p q , q is firm i’s first period duopoly expected profits. Let,
1 1
1 D
1 1
2 2
1 2
p Q 5p q , q 1p q , q , be the duopoly first period profits when firm i
1 1
1 1
1 1
1 1
10
See Eq. 9.
176 M
. Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
i 1
2 D
ˆ ˆ
chooses q , i 51,2, and where Q 5 q 1q , and let V uQ 52VuQ , be the
1 1
1 1
1 1
duopoly second period expected profits. The next Proposition gives us a sufficient condition to compare the monopoly and duopoly experimental behavior:
Proposition 4. The monopoly will experiment at least as much as the duopoly , i.e.,
M M
11
ˆ ¯
ˆ ¯
ˆ Q 2 Q Q 2 Q
whenever E[VuQ ] is concave in Q.
1 1
1 1
M i
ˆ Proof. First, let us show that Q 1a 6m. q , i 51, 2. Indeed, since the first period
1 1
ˆ demand is P 5a 2uQ 1´, then E[P]5a 2mQ and hence a 2mQ 0. Therefore, Q a
1 1
M M
M M
1
¯ ¯
ˆ ˆ
ˆ
]
m 52Q . By lemma 3, Q Q . Thus, Q Q . This implies that
1 1
1 1
1 2
a
M i
ˆ ˆ
] q , Q
1 27
1 1
6m
M M
M M
M
ˆ ¯
ˆ ¯
ˆ ˆ
¯ ˆ
ˆ Next, assume that Q 2 Q . Q 2 Q . Then, Q . Q 1 Q 2 Q 5 Q 1a 6m,
1 1
1 1
1 1
1 1
1
where the last equality comes from Eq. 24 and the myopic monopoly output. Clearly,
M
ˆ for Q 5a 6m the above inequality is not feasible. Hence, w.l.o.g assume that
1 M
M
ˆ ˆ
Q 5a 6m. Since Q is optimal for the monopoly,
1 1
a q
M M
M M
M M
M
ˆ ˆ ˆ
ˆ ˆ
ˆ ]
] p
Q 1 E
f
V uQ
g
p
S
Q 2
D
1 E
F
V
S
u
S
Q 2
DDG
1 1
1 1
1 1
6m 6m
28
i i
2
ˆ ˆ
ˆ ˆ
and since q 51 2Q and q , q is the first period Nash equilibrium output,
1 1
1 1
a
M 1
1 2
D 2
2
ˆ ˆ ˆ
ˆ ˆ
] ˆ
ˆ p q , q 1 E VuQ p
S
Q 1
2 q , q
D
f g
1 1
1 1
1 1
1 1
6m q
M M
ˆ ˆ
] 1 E
F
V
S
u
S
Q 2
DDG
29
1
6m a
M 2
1 2
D 1
1
ˆ ˆ ˆ
ˆ ˆ
] ˆ
ˆ p q , q 1 E VuQ p
S
Q 1
2 q , q
D
f g
1 1
1 1
2 1
1 1
6m q
M M
ˆ ˆ
] 1 E
F
V
S
u
S
Q 2
DDG
30
1
6m Observe that by Eq. 27, inequalities Eqs. 29 and 30 are well defined.
Adding up these two inequalities we obtain: a
q
M D
D D
D
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
] ]
p Q 1 E V uQ p
S
Q 1
,Q
D
1 E
F
V
S
u
S
Q 2
DDG
f g
1 1
1 1
1 1
6m 6m
31 Eqs. 28 and 31 can be rewritten as follows
a a
M M
M M
M M
ˆ ˆ
ˆ ˆ
ˆ ˆ ]
] p
Q 2 p
S
Q 2
D
E
F
V
S
u
S
Q 2
DDG
2 E
f
V uQ
g
32
1 1
1 1
1 1
6m 6m
11
Notice that under this situation a first period duopoly equilibrium in pure strategies will always exist, i.e. Eq.
i
13 is continuous and strictly concave in q .
1
M . Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
177
a a
M M
D D
D D
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
] ]
E V uQ 2 E
F
V
S
u
S
Q 1
DDG
p
S
Q 1
, Q
D
2 p Q
f g
1 1
1 1
1 1
1
6m 6m
33 To derive a contradiction it is sufficient to show that, by Eqs. 32 and 33:
a
M M
M M
D D
ˆ ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ
] E
F
V
S
u
S
Q 2
DDG
2 E
f
V uQ
g
. E V uQ 2 E
F
V
S
u
S
Q
f g
1 1
1 1
6m a
] 1
DDG
A 6m
and a
a
M M
M M
D D
ˆ ˆ
ˆ ˆ
ˆ ]
] p
Q 2 p
S
Q 2
D
p
S
Q 1
, Q
D
2 p Q
B
1 1
1 1
1 1
1 1
1
6m 6m
Let us first prove A.
M
ˆ Notice that if E[V uQ ] is concave in Q , then
1 1
a
M M
M M
ˆ ˆ
ˆ ˆ ˆ ˆ
] E
F
V
S
u
S
Q 2
DDG
2 E
f
V uQ
g
E V uQ
f g
1 1
1
6m a
M M
ˆ ˆ
] 2 E
F
V
S
u
S
Q 1
DDG
34
1
6m
M M
ˆ Next, let P 5a 2uQ 11´, P 5a 2uQ 1a 6m1´ and let
1 1
1
ˆ mh 1 tQ a 2 P
1 1
ˆ ]]]]]
u 5
2
ˆ h 1 tQ
1
and a
M M
ˆ ]
mh 1 t
S
Q 1
D
a 2 P
1 1
M
6m ˆ
]]]]]]]]] u
5 .
2
a
M
ˆ ]
h 1 t
S
Q 1
D
1
6m
M 2
D 2
ˆ ˆ
ˆ ˆ
Also recall that, V u 5a 4u , V 52Vu 52a 9u , then define: a
M M
M M
D
ˆ ˆ ˆ
ˆ ]
W 5 E V uQ 2 E
F
V
S
u
S
Q 1
DDG
5
f g
1 1
6m
M M
M M
M
ˆ ˆ
E
V u fa 2 P dP 2
E
V u fa 2 P dP 5
1 1
1 1
2
a 1
1
M M
] ]
]
E
fa 2 P dP 2
E
fa 2 P dP . 0
F G
1 1
1 1
M
4 ˆ
ˆ u
u
M
ˆ ˆ
by the proof of lemma 3, since Q . Q 1a 6m. Also, define:
1 1
a
M M
D D
D
ˆ ˆ ˆ
ˆ ]
W 5 E V uQ 2 E
F
V
S
u
S
Q 1
DDG
5
f g
1 1
6m
178 M
. Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
M D
D M
M
ˆ ˆ
E
V u fa 2 P dP 2
E
V u fa 2 P dP 5
1 1
1 1
2
2a 1
1
M M
] ]
]
E
fa 2 P dP 2
E
fa 2 P dP . 0
F G
1 1
1 1
M
9 ˆ
ˆ u
u again by lemma 3. Hence
2
1 1
a 2a2
M D
M M
F G
] ]
] ]]
W 2 W 5
E
fa 2 P dP 2
E
fa 2 P dP 2
F G
1 1
1 1
M
ˆ ˆ
4 9
u u
. 0 35
and by Eqs. 34 and 35, A is established. In Appendix A we prove B and then, the result of the Proposition follows.
j As we noted above, given our normality assumptions, the signal and the mean of the
ˆ updated beliefs, u, may take on negative values. However, the probability of such an
event can be made arbitrarily small by appropriately choosing the variances of the model.
¯ ˆ
Let P 5a 1mh tQ be the price which makes posterior beliefs u 50, then the
1
ˆ probability of u ,0 is arbitrarily small whenever it is small the probability of any price
˜ ˜
¯ ¯
observation P .P. In particular, notice that P 5E[P ]1varP mh Q and hence
1 12
˜ ˜
¯ ProbP .P 5ProbP 2E[P ].varP mh Q which by Chebyshev’s Inequality,
by
1
˜ ˜
the definition of varP and for varP mh Q .0 has the following upper bound:
1 2
tQ mh
1
ˆ ]
]] Prob P 2 E[P ] . varP
, .
S D
2
Q ˆ
1
m h This upper bound decreases with h and m, i.e. with a more precise and with a higher
˜ mean random slope u. Obviously, if h is big enough we can make the above probability
˜ ˜
as small as we want. Also, if h is high, the market signal P or a 2P Q is quite
1 13
precise as well since any of their variances decreases in h. Hence, what we need is a
˜ random variable u not too ‘diffuse’, which will make, in turn, a market signal not too
‘diffuse’ as well. ˆ
ˆ ˆ
Let P5a 1mh tQ 3h 22h 3h 1h be the price at which u 53mh 3h 1
1
˜ ˆ
ˆ ˆ
¯ h .0 or h 1h 3u 5mh, and with P,P. The next lemma shows that if u is
ˆ precise enough, then E[VuQ ] is concave in Q.
ˆ
Lemma 5. A sufficient condition for E[VuQ ] to be concave in Q is that the variance
12
Chebyshev’s Inequality: Let X be a random variable, and d .0. Then VarX
]] ProbX 2E[X].d ,
2 13
2 2
d ˜
˜ ˆ
˜ Recall that P 5a 2uQ 1´, and then varP 5Q h11 t 5tQ ht5h ht where h and t are the
˜ ˜
˜ precisions of u and ´ respectively. Note that varP is a decreasing function of both h and t. Also, recall that,
2 2
for any Q , the variance of the market signal is tQ 1h htQ .
1 1
1
M . Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
179
˜ of the random demand
’s parameter u be sufficiently small to make the probability of any price observation P .P small enough
. ¯
Moreover if that variance is such that the distance , P2P is not too big, then making
ˆ ˆ
the probability of u ,0 arbitrarily small guarantees the concavity of E[VuQ ] in Q .
Proof. In Appendix A it is shown that:
2
ˆ ˆ
≠ E[VuQ ] 1
h
1
ˆ ˆ
ˆ
SF G D
]]]] ]]
] 5
E
V -u 2 ≠u ≠P h 1
u 2 mh fa
f g
2
F
1
G
2
ˆ 3
≠Q h Q
1 1
2 P dP 5
1 1
ˆ t
u ˆ
ˆ
SF G D
] ]
E
V -u h 1
u 2 mh fa 2 P dP
1 1
3
ˆ 3
h ˆ
ˆ Notice that V -u is a negative, non-increasing function of P , i.e. ≠V -u ≠P 5
1 1
ˆ ˆ
ˆ V +du dP 0 since V + . 0. Also, [h 1h 3]u 2mh is a linear, non-increasing
1
function of P , positive for P P and negative for P .P at the mean of P , i.e. at
1 1
ˆ P 5a 2mQ , this function is hm 3.0.
1 1
Hence the above expression is negative for P [2`, P and positive for P [P,
1 1
ˆ ` notice that it is negative for all P if u is constant in P . Therefore if the probability
1 1
of P .P is small enough, then the whole expression is nonpositive and hence
1
ˆ E[VuQ ] is concave in Q. Again, by Chebyshev’s Inequality and by the definition of
14
P,
2 2
ˆ ˆ
tQ 3h 1 h mhh
1
˜ ˜ ]]]]
]]]] Prob P 2 E[P ] . varP
, .
S D
2
ˆ ˆ
Q 3h 1 h m hh
1
where again this upper bound decreases with h and m.
2
¯ ˆ
ˆ Also, notice that if the distance, P 2P53mhh tQ 3h 1h 53mh Q 3h 1
1 1
˜ ˆ
¯ h varP .0 is not too big, it is only needed that the probability of P .P is small
ˆ enough to guarantee the concavity of E[VuQ ].
j
5. Conclusions
