M . Dolores Alepuz, A. Urbano Mathematical Social Sciences 37 1999 165 –188
173
2
ˆ 4a tQ
2a 1
2a ]
1
ˆ ]
]] ]
] Q 5
1
E
fa 2 P dP . 5Q j
1 1
1 1
2 3
3m ˆ
ˆ 3m
27h m u
A result similar to Proposition 1 was first obtained by Grossman et al., 1977. However, the basic idea behind the proof of our result is based on the notion of ‘more
informative signals’ while that of GKM is it on ‘sufficient experiments’. Nevertheless, by Blackwell, 1951 Theorem one can go from one direction to the other. The same
findings can be obtained under the Monotone Likelihood Ratio Property MLRP as in
ˆ Mirman et al., 1993a and Mirman et al., 1994. The intuitive explanation is that if Q .Q,
ˆ then the information structure induced by Q is sufficient for that induced by Q in the
sense of Blackwell, thus making price by Blackwell’s theorem a more informative signal. To the best of our knowledge, no paper in market experimentation has shown yet
directly that higher quantities produce more informative market signals. We are able to prove it since we deal with the closed form of the value function and the random
variables are normally distributed. This is the aim of the next section.
3. More informative signals
As we said above, the reason why each firm’s output under experimentation is larger than its corresponding myopic output, is that firms produce more to gather information
about Nature’s choice of u. Let us be more specific. Observe that in the second period ˆ
ˆ the posterior belief about u is a Normal distribution with mean u and precision h.
1 2
ˆ ˆ
ˆ However, in the first period, before q and q have been produced, u is itself a random
1 1
variable that is generated by the signal of the parameter u, i.e. a 2P Q . Let
1 1
S 5a2P Q and let us make the following definition.
Q 1 1
1
Definition 2. A random variable X is more informative than a random variable Y if the distribution of X dominates that of Y in the sense of second order stochastic dominance.
Let x and x be any two values of the first period total output, and let Sx and Sx be
1 2
1 2
the corresponding signals – random variables – of the parameter u, generated by these outputs.
Observe that by Eq. 5, for any value x of the total output,
2
htx ]]
SxN m, 25
S D
2
h 1 tx ˆ
Moreover, by Eq. 7, these signals Sx and Sx lead to some posterior about u, uSx
1 2
1
ˆ and uSx , respectively.
2 2
2
ˆ ˆ
ˆ ˆ
Define ux, s5mh 1tx s h 1tx and notice that u 5uQ , S for u as in Eq. 7.
1 Q
1
The next lemma shows that the precision of a signal and hence its informative content increases in x.
Lemma 3. Sx is more informative than Sx if :
1 2
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
