International Journal of Industrial Organization
19 (2001) 1563–1582
www.elsevier.com / locate / econbase
Price and quantity experimentation: A synthesis
Paul Belleflamme a , *, Francis Bloch b
a
Department of Economics, Queen Mary and Westfield College, University of London, London, UK
b
Department of Economics, IRES, Universite´ catholique de Louvain, Louvain, Belgium
Received 27 January 1998; received in revised form 1 March 1999; accepted 27 December 1999
Abstract
This paper compares experimentation about product differentiation in a linear setting
under four market structures: quantity-setting and price-setting monopoly, Cournot and
Bertrand duopoly. Quantity-setting firms always experiment by raising their quantities and
the monopolist experiments relatively more than the duopolists. A price-setting monopolist
does not experiment. The value of information to Bertrand duopolists may be positive or
negative depending on the degree of product differentiation. When information is valuable,
price-setting duopolists experiment by lowering prices. A numerical example indicates that
the intensity of experimentation is higher in a Cournot duopoly than in a Bertrand duopoly.
 2001 Elsevier Science B.V. All rights reserved.
JEL classification: D83; L13
1. Introduction
Firms facing uncertain demand can acquire information by experimenting:
setting prices or quantities away from the short run equilibrium in order to learn
the true value of demand in the long run. Recent studies of the dynamic behavior
of firms facing uncertain demand have pointed out that the firms’ incentives to
acquire information and the direction in which experimentation affects prices and
* Corresponding author. Tel.: 144-20-7882-5587; fax: 144-20-8983-3580.
E-mail address: [email protected] (P. Belleflamme).
0167-7187 / 01 / $ – see front matter  2001 Elsevier Science B.V. All rights reserved.
PII: S0167-7187( 00 )00062-X
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quantities depend greatly on the assumptions made on market competition. In a
monopoly with homogeneous products, Mirman et al. (1993) show that pricesetting and quantity-setting firms adopt different behavior. Mirman et al. (1994)
and Alepuz and Urbano (1999) extend the analysis to a Cournot duopoly and note
that the incentives to acquire information are different in a monopoly and a
duopoly. Using different models of product differentiation, Aghion et al. (1993),
Alepuz and Urbano (1994) and Harrington (1995) have emphasized that the firms’
incentives to acquire information about product differentiation leads to price
dispersion in equilibrium.
While these studies give a clear understanding of the incentives of firms to
experiment and of the key variables affecting the direction of experimentation,
they do not provide an easy framework to compare learning and experimentation
in different market structures. Is the direction of experimentation identical in
models of price and quantity competition? How is the magnitude of experimentation affected by the type of competition on the market? Is experimentation higher
in a monopoly or a duopoly? Our objective in this paper is to shed light on these
questions, by studying learning and experimentation in a simple, unified model of
market competition.
We consider a market with two differentiated products which allows for a
simple comparison between Bertrand and Cournot competition. As in Aghion et al.
(1993), Alepuz and Urbano (1994) and Harrington (1995), we suppose that the
firms are uncertain about the degree of differentiation of the two products. We
consider a two-period model of competition. In the first period, firms set prices and
quantities. They observe the market outcome and update their beliefs about the
degree of product differentiation, and compete again on the market in the second
period. We assume the following simple informational structure. The degree of
product differentiation can only take on two values: one where the products are
independent, one where the products are substitutes. The firms do not observe the
market outcome perfectly, since the representative consumer’s utility function is
subject to random shocks drawn from a uniform distribution.
In this specific model, we are able to compute exactly the equilibria of the
two-period game. We show that a quantity-setting monopolist and Cournot
duopolists always experiment by raising their quantities. We also prove that
information is not valuable to a price-setting monopolist and, as in Harrington
(1995), that the value of information to Bertrand duopolists may be positive or
negative according to the degree of differentiation of the products. If the products
are close substitutes, information is valuable; if, on the other hand, products are
highly differentiated, the value of information is negative. When information is
valuable, Bertrand duopolists lower their prices in order to acquire information;
the intensity of experimentation is thus higher in the duopoly than in the
monopoly. When information has a negative value, Bertrand duopolists raise their
prices in order to limit information acquisition. We further show that the intensity
of experimentation is higher in the quantity-setting monopoly than in the quantity-
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1565
setting duopoly according to two different measures: the relative deviation from
the myopic equilibrium of the probability to learn the truth and of the first-period
quantities. Furthermore, we establish that the loss in first-period profits is smaller
in the monopoly than in the duopoly. Finally, we provide a numerical example to
compare the magnitude of experimentation in the Cournot and Bertrand duopolies
(in the case where information is valuable for the Bertrand duopolists). This
example suggests that the intensity of experimentation is higher in the Cournot
model. Both the deviation in the probability to learn the truth and the first-period
losses are higher for quantity-setting firms.
In a recent paper, Alepuz and Urbano (1999) also compare the incentives to
experiment in a monopoly and a Cournot duopoly. They find that the monopoly
experiments more than the duopoly when the demand’s unknown parameter is
sufficiently precise. This result stems from the fact that, when information is
precise, Cournot competitors may be reluctant to transmit information to each
other, since more accurate information strengthens competition on the market. As
Alepuz and Urbano (1999) consider a homogeneous market where the scale
parameter of demand is unknown while we analyze a market with differentiated
products with unknown degree of product differentiation, their results are not
directly comparable to ours.
The rest of the paper is organized as follows. We introduce the basic model in
Section 2. In Section 3, we analyze experimentation when firms set quantities. In
Section 4, we analyze experimentation when firms set prices. We compare Cournot
and Bertrand duopolies through a numerical example in Section 5, and conclude in
the final Section.
2. The model
We consider a differentiated products industry with two goods, denoted A and
B. The demand side of the market is represented by a representative consumer
with the utility function
U(qA , qB ) 5 (a 1 eA )qA 1 (a 1 eB )qB 2 (1 / 2)(q 2A 1 q 2B 1 2g qA qB ).
The parameter a (a . 0) measures the absolute size of the market; eA and eB are
independent random shocks affecting the taste parameters of products A and B; the
parameter g (0 < g < 1) is an indicator of the degree of substitutability between
the two goods.
The consumer’s maximization problem yields the linear inverse demand
schedule
pi 5 a 2 qi 2 g q j 1 ei ,
in the region of quantities where prices are positive. The demand schedule, for
g ± 1, is then given by
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qi 5 a 2 b ( pi 2 ei ) 1 h( pj 2 ej ),
in the region of prices where quantities are positive, with
a 5 a /(1 1 g ), b 5 1 /(1 2 g 2 ), h 5 g /(1 2 g 2 ).
We suppose that the producing firm(s) are uncertain about the degree of product
differentiation, g. To keep the analysis tractable, we assume that g can only take
on two values: either g 5 0 (the goods are independent), or g 5 g¯ , with 0 , g¯ , 1
(the goods are substitutes). The prior probability that the goods are substitutes is
given by r0 . We further assume that the random shocks are uniformly distributed
over [2e, e ], with e > [(a 2 c) g¯ ] /(1 2 g¯ ).
We analyze the effects of learning and experimentation in a two-period model.
In the first period, firm(s) set prices or quantities and observe the realization of
demand or inverse demand. On the basis of this information, firm(s) update their
beliefs about the degree of product differentiation g and select prices (or
quantities) accordingly in the second period.
We consider four possible market configurations. Either the two goods are
produced by the same firm (monopoly) or by different firms (duopoly). In each
market structure, we analyze both the case of price-setting (Bertrand) and
quantity-setting (Cournot) firms. The production technology is characterized by a
constant marginal cost of production c. The firm(s) maximize the discounted sum
of profits in the two periods. For a monopoly, we derive the optimal choice of
prices or quantities for the two periods. For a duopoly, we analyze a multi-stage
game with observed actions and symmetric incomplete information. Accordingly,
the equilibrium concept we use is a Bayesian subgame perfect equilibrium. In both
instances, we proceed by backward induction following a three-step procedure.
Step 1. We solve for the second period profit as a function of the posterior beliefs
r held by the firm(s) on the value of the degree g of product differentiation; we
denote it by V( r ).
Step 2. Given the informational structure of the model, posterior beliefs can only
take three values. Either firm(s) do not learn anything and keep their prior beliefs,
or they learn with certainty the true value of g. We compute the probability that
firm(s) do not acquire any information about g as a function of the strategic
variables, sA and s B , chosen in the first period; we denote it by W(sA , s B ).
Step 3. We express the expected profit of the firm as a function of the first period
variables:
P (sA , s B ) 5 P1 (sA , s B )
1 dh[1 2 W(sA , s B )][ r0V(1) 1 (1 2 r0 )V(0)] 1 W(sA , s B )V( r0 )j
and compute the optimal values of sA and s B .
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1567
3. Quantity setting firms
3.1. Monopoly
Step 1. We compute the second period profit, when the monopolist holds a belief
r. The monopolist’s profit is given by
P (qA , qB ) 5 qA (a 2 qA 2 rg¯ qB 2 c) 1 qB (a 2 qB 2 rg¯ qA 2 c),
yielding an optimal solution
a2c
qA 5 qB 5 ]]]
2(1 1 rg¯ )
(1)
and an optimal profit
(a 2 c)2
VM ( r ) 5 ]]].
2(1 1 rg¯ )
(2)
Since the profit is a convex function of the beliefs r, information is valuable to
the monopolist. (See Mirman et al., 1993).
Step 2. We now compute the posterior beliefs as a function of the quantities
chosen in the first period. We first express the conditions under which a firm will
not infer any information from observing prices and quantities of the two goods.
This happens whenever the realizations of the random shocks eA and eB fall inside
the interval [2e, e ] for the two possible values of g. That is,
pA 2 a 1 qA 1 g¯ qB < e
(3)
pA 2 a 1 qA > 2 e
(4)
pB 2 a 1 qB 1 g¯ qA < e
(5)
pB 2 a 1 qB > 2 e.
(6)
We now compute, for any choice of qA , qB , the ex ante probability that the four
preceding conditions are met. We first suppose that g 5 g¯ . In that case,
pA 5 a 2 qA 2 g¯ qB 1 eA
pB 5 a 2 qB 2 g¯ qA 1 eB .
It is easily seen that conditions (3) and (5) are always met, while conditions (4)
and (6) give
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eA > g¯ qB 2 e
eB > g¯ qA 2 e.
Since e > [(a 2 c) g¯ ] /(1 2 g¯ ), qA < (a 2 c), qB < (a 2 c) and g¯ , 1, we have
that 2e . g¯ qA and 2e . g¯ qB for all qA , qB . Hence, we can compute the ex ante
probability that the firm does not learn the value of g as
prob( r 5 r0 ug 5 g¯ ) 5 (2e 2 g¯ qA )(2e 2 g¯ qB ) /(4e 2 ).
Proceeding in the same way when g 5 0, we also obtain
prob( r 5 r0 ug 5 0) 5 (2e 2 g¯ qA )(2e 2 g¯ qB ) /(4e 2 ).
Hence,
W(qA , qB ) 5 prob( r 5 r0 ) 5 (2e 2 g¯ qA )(2e 2 g¯ qB ) /(4e 2 ).
(7)
In a quantity-setting framework, firms thus acquire information by raising their
quantities.
Step 3. We now turn to the optimal choice of the monopolist in the first period.
The expected profit of the monopolist over the two periods is given by
EPM 5 qA (a 2 qA 2 r0 g¯ qB 2 c) 1 qB (a 2 qB 2 r0 g¯ qA 2 c)
1 d W(qA , qB )VM ( r0 ) 1 d [1 2W(qA , qB )][ r0VM (1) 1 (1 2 r0 )VM (0)].
Straightforward but tedious computations give the optimal value of qA , qB as
qA 5 qB 5 q EM
2e [ 2e (a 2 c) 1 dg¯ [ r0VM (1) 1 (1 2 r0 )VM (0) 2VM ( r0 )] ]
5 ]]]]]]]]]]]]]]]]
.
8e 2 (1 1 r0 g¯ ) 1 dg¯ 2 [ r0VM (1) 1 (1 2 r0 )VM (0) 2VM ( r0 )]
(8)
To compare the optimal quantity of the experimenting monopolist with the
optimal quantity of the myopic monopolist, we recall that the myopic monopolist
chooses
a2c
q NE
.
M 5 ]]]]
2(1 1 r0 g¯ )
A simple derivation shows
q EM 2 q NE
M
dg¯ [ r0VM (1) 1 (1 2 r0 )VM (0) 2VM ( r0 )][4e (1 1 r0 g¯ ) 2 g¯ (a 2 c)]
5 ]]]]]]]]]]]]]]]]]]]]
2(1 1 r0 g¯ ) [ 8e 2 (1 1 r0 g¯ ) 1 dg¯ 2 [ r0VM (1) 1 (1 2 r0 )VM (0) 2VM ( r0 )] ]
. 0.
(9)
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1569
Proposition 1. The quantity-setting monopolist experiments by increasing the
quantities of the two goods.
3.2. Duopoly
Step 1. As before, we first compute the equilibrium profit in the second period,
when each firm holds a belief r. Firm i chooses its quantity qi to maximize its
second period profit
Pi (qi , q j ) 5 qi (a 2 qi 2 rg¯ q j 2 c).
The equilibrium quantities and profits are easily computed as
a2c
qA 5 qB 5 ]]
2 1 rg¯
(10)
(a 2 c)2
VD ( r ) 5 ]]]2 .
(2 1 rg¯ )
(11)
Since the second period profit is a convex function of r, the value of information is
always positive in the Cournot duopoly.
Step 2. The construction of the ex ante probability to learn the value of g is
identical in the Cournot duopoly and the quantity-setting monopoly.
Step 3. In the first period, firm i’s expected profit over the two periods is given
by
EPD,i 5 qi (a 2 qi 2 r0 g¯ q j 2 c)
1 d W(qA , qB )VD ( r0 ) 1 d [1 2 W(qA , qB )][ r0VD (1) 1 (1 2 r0 )VD (0)].
The symmetric Nash equilibrium is given by
qA 5 qB 5 q DE
2e [ 2e (a 2 c) 1 dg¯ [ r0VD (1) 1 (1 2 r0 )VD (0) 2VD ( r0 )] ]
5 ]]]]]]]]]]]]]]]]
.
4e 2 (2 1 r0 g¯ ) 1 dg¯ 2 [ r0VD (1) 1 (1 2 r0 )VD (0) 2VD ( r0 )]
Recall that the myopic Nash equilibrium is given by
a2c
qA 5 qB 5 q DNE 5 ]]]
.
2 1 r0 g¯
We can thus compute
(12)
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1570
q ED 2 q NE
D
dg¯ [ r0VD (1) 1 (1 2 r0 )VD (0) 2VD ( r0 )][2e (2 1 r0 g¯ ) 2 g¯ (a 2 c)]
5 ]]]]]]]]]]]]]]]]]]]
(2 1 r0 g¯ ) [ 4e 2 (2 1 r0 g¯ ) 1 dg¯ 2 [ r0VD (1) 1 (1 2 r0 )VD (0) 2VD ( r0 )] ]
. 0.
(13)
Proposition 2. In a Cournot duopoly, firms experiment by increasing their
quantities.
Propositions 1 and 2 are easy to interpret. In the quantity-setting case, the value
of information is always positive. Since firm(s) acquire information by raising their
quantities, experimenting firm(s) produce more than in the myopic equilibrium.
3.3. Comparison
There is no obvious indicator of the intensity of experimentation. We use three
different measures which are expressed in terms of relative changes from the
myopic optimum. The first indicator, Mr , measures the change in the probability to
learn the true value of the unknown parameter, the second indicator, Mp , the
change in first-period profits, and the last indicator, Mq , the change in first-period
quantities. Formally,
f1 2 W(q E, q E )g 2f1 2 W(q NE, q NE )g
Mr 5 ]]]]]]]]]]
,
1 2 W(q NE , q NE )
p E1 2 p NE
1
]]]
Mp 5
,
p NE
1
E
NE
q 2q
Mq 5 ]]]
.
q NE
Proposition 3. The intensity of experimentation is higher in the quantity-setting
monopoly than in the quantity-setting duopoly according to measures Mr and Mq .
The loss in first-period profits is smaller in the monopoly than in the duopoly.
D
M
D
Proof. We need to prove that (i) M M
q . M q . 0, (ii) 0 . M p . M p , and (iii)
M
D
M r . M r . 0.
Proof of part (i). We introduce the following notation:
M ; r0VM (1) 1 (1 2 r0 )VM (0) 2VM ( r0 ),
D ; r0VD (1) 1 (1 2 r0 )VD (0) 2VD ( r0 ).
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1571
We first show that M . D. Define h(g ) 5 1 /(1 1 g ) 2 1 /(2 1 g )2 . Simple
computations show that h is a convex function of g. Therefore, by Jensen’s
inequality, we obtain M . D. Next, from expressions (9) and (13), we can rewrite
D
the condition for M M
q . M q as (with b ; a 2 c)
2(1 1 r0 g¯ )(2´b 1 dg¯ M) (2 1 r0 g¯ )(2´b 1 dg¯ D)
]]]]]]]
. ]]]]]]]
.
8´ 2 (1 1 r0 g¯ ) 1 dg¯ 2 M
4´ 2 (2 1 r0 g¯ ) 1 dg¯ 2 D
Developing the latter inequality, we have an equivalent expression:
2´(2 1 r0 g¯ )f4´(1 1 r0 g¯ ) 2 b g¯ g(M 2 D) 1 r0 g¯ 2 D(2´b 1 dg¯ M) . 0.
Since ´ > (b g¯ ) /(1 2 g¯ ) and M . D, the first term is positive and the inequality
is thus satisfied. Moreover, we know from Proposition 2 that M Dq . 0.
E
To prove parts (ii) and (iii) we need first to establish that q DE . q M
. Tedious but
E
straightforward computations show that the sign of the difference s q NE
D 2 q Md is
equal to the sign of the following polynomial of the second degree in b ; a 2 c:
dg¯ 3 (1 2 r0 )b 2 2 2d´g¯ 2 (1 2 r0 )(2 1 r0 g¯ )b 1 8´ 2 (1 1 g¯ )(1 1 r0 g¯ ).
It is readily established that this polynomial admits no real root and is thus
E
E
positive for all b. Since, from Proposition 2, q ED . q NE
D , we have that q D . q M .
Proof of part (ii). Using the definition of first-period profits and regrouping
terms, we can express the relative changes in first-period profits alternatively as:
F
F
G
G
E
(1 1 r0 g¯ )q M
M
M
M p 5 M q 1 2 ]]]]]
NE ,
b 2 (1 1 r0 g¯ )q M
(1 1 r0 g¯ )q DE
D
D
M p 5 M q 1 2 ]]]]]
.
b 2 (1 1 r0 g¯ )q DNE
M
D
First, it is obvious that both M p and M p are negative since first-period profits
are maximized when the myopic quantities are produced. Second, we know from
D
part (i) that M M
q . M q . Furthermore, combining the result we have just shown and
the results from Propositions 1 and 2, we have the following ranking of quantities:
E
M
D
q ED . q DNE . q M
. q NE
M . It is thus clear that M p . M p .
Proof of part (iii). The proof follows the same lines as in (ii). The relative
changes in the probability to learn the truth are alternatively expressed as:
F
F
G
G
ḡq EM
M
M
M r 5 M q 1 2 ]]]
NE ,
4´ 2 g¯ q M
ḡq DE
D
D
M r 5 M q 1 2 ]]]
.
4´ 2 g¯ q DNE
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D
We then make use of the previous results to establish that M M
r . M r . Finally,
NE
E
D
2´ 2 g¯ q D . 2´ 2 g¯ q D . 0 implies that M r . 0. h
Proposition 3 shows that the intensity of experimentation, as measured by
deviations in the quantities and in the probability to learn the truth, is higher in a
monopoly than in a duopoly when quantity is the strategic variable. The intuition
underlying this result can be grasped as follows. Consider the effect of an increase
in quantity above the myopic equilibrium on the profits of the monopolist and of
the duopolist producing good A. By the envelope theorem, this increase has a
negligible effect on first-period profits. However, the effect on expected secondperiod profits is very different for a monopolist or a duopolist. On the one hand, as
shown in the proof of Proposition 3, the value of information is larger for a
monopolist than for a duopolist (i.e., M . D). On the other hand, the effect of an
increase in quantities on the probability to learn the truth is higher for a
monopolist than for a duopolist, since the myopic monopoly quantities are lower
than the duopoly myopic quantities and W( ? ) is decreasing in qA and qB . Both
effects concur to make the marginal effect of an increase in quantities on expected
second-period profits higher for a monopoly than for a duopoly. Proposition 3 also
shows that the loss in first-period profits is relatively lower for the monopoly than
for the duopoly. To understand this effect, note that around the myopic equilibrium, the increase in quantities due to experimentation has a negligible effect on
the monopolist’s first-period profits. For the duopolists, however, an increase in
the other firm’s quantity has an adverse effect on profits. Hence, locally around the
myopic equilibrium, the loss in first-period profits due to experimentation is higher
for the duopolists than for the monopolist.1
4. Price setting firms
4.1. Monopoly
In the second period, the price-setting monopoly holding beliefs r chooses
prices pA and pB to maximize its profit
P ( pA , pB ) 5 ( pA 2 c)[a ( r ) 2 b ( r )pA 1 h( r )pB ]
1 ( pB 2 c)[a ( r ) 2 b ( r )pB 1 h( r )pA ],
where a ( r ), b ( r ) and h( r ) are the expected values of the parameters a, b and h,
i.e.,
ra
a ( r ) 5 ]]
1 (1 2 r )a,
1 1 g¯
1
r
b ( r ) 5 ]]2 1 1 2 r,
1 2 g¯
We thank an anonymous referee for providing us with this intuition.
rg¯
h( r ) 5 ]]2 .
1 2 g¯
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1573
The optimal prices pA and pB are given by
pA 5 pB 5 (a 1 c) / 2.
In our model, the price-setting monopolist chooses prices which are independent
of the degree of product differentiation, and hence of the beliefs r. This striking
result stems from the specific functional form of demand in our model. In fact we
may rewrite demand for good i as a function of the price pi and the price
dispersion pj 2 pi :
qi 5 a ( r ) 2f b ( r ) 2 h( r )g pi 1 h( r )s pj 2 pid
r
5 ]]] 1 (1 2 r )
(1 1 g¯ )
F
rg¯
p 2 p dG.
GFa 2 p 1 ]]]]]]
(1 2 g¯ )s1 1 g¯ 2 rg¯ d s
i
j
i
As its profit is symmetric in the two products, the monopolist will select pi 5 pj .
But when price dispersion is equal to zero, the profit on market i can be written as
Pi 5 ( pi 2 c)sa 2 pidf r /(1 1 g¯ ) 1 (1 2 r )g so that the monopolist’s optimal price
is independent of g¯ . Hence, the monopolist’s optimal profit is a linear function of
r, information has no value and no experimentation occurs.2 We record this fact in
the following proposition.
Proposition 4. There is no experimentation in the price-setting monopoly. The
monopolistic firm optimally selects two prices pA and pB which are independent of
the beliefs.
4.2. Duopoly
Step 1. In the second period, when firms hold beliefs r, each firm’s profit is
given by
Pi ( pi , pj ) 5 ( pi 2 c)[a ( r ) 2 b ( r )pi 1 h( r )pj ].
The unique Nash equilibrium of the game is then computed as
a ( r ) 1 b ( r )c r (1 2 g¯ )a 1 (1 2 r )(1 2 g¯ 2 )(a 1 c)
pA 5 pB 5 ]]]] 5 ]]]]]]]]]]
,
2b ( r ) 2 h( r )
2 2 rg¯ 2 2(1 2 r ) g¯ 2
yielding an equilibrium profit
2
Other specifications of demand could lead the price setting monopolist to experiment as in Mirman
et al. (1993).
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F
b ( r ) 2 h( r )
VB ( r ) 5 b ( r ) ]]]]
2b ( r ) 2 h( r )
G
2
(a 2 c)2
(1 2 g¯ ) [1 1 r (1 2 g¯ 2 )] [1 1 (1 2 r ) g¯ ] 2
5 ]]] ]]]]]]]]]
(a 2 c)2 .
2 2
(1 1 g¯ )
¯
¯
[2 2 rg 2 2(1 2 r ) g ]
Contrary to the case of quantity-setting firms, the profit is not necessarily a
convex function of beliefs. In fact, as in Harrington (1995) and Alepuz and
Urbano (1994), information may or may not be valuable, depending on the values
of the parameters. In our simple model, information has positive value if
r0VB (1) 1 (1 2 r0 )VB (0) 2VB ( r0 ) . 0 and negative value if r0VB (1) 1 (1 2
r0 )VB (0) 2VB ( r0 ) , 0. Computations show that
4(1 1 g¯ )(1 2 g¯ )2
]]]]]
information has positive value if r0 .
,
g¯ (4 2 7 g¯ 1 4 g¯ 2 )
4(1 1 g¯ )(1 2 g¯ )2
information has negative value if r0 , ]]]]]
.
g¯ (4 2 7 g¯ 1 4 g¯ 2 )
(14)
Notice that the right hand side of these inequalities defines a monotonically
decreasing function of g¯ . Hence, the more the products are differentiated (the
lower the value of g¯ ), the more likely it is that information has a negative value.
In fact, as long as g¯ , ]23 , information always has a negative value. If g¯ . ]23 ,
information may have a positive value if initial beliefs r0 are high enough. Thus,
in accordance with Harrington (1995), we find that information has a positive
value when the products are likely to be highly substitutable and a negative value
when the products are likely to be highly differentiated.
Step 2. We now compute the posterior beliefs as a function of the prices chosen
in the first period. Recall that the firm does not learn the value of g whenever the
four inequalities (3)–(6) are satisfied. Suppose first that g 5 g¯ . In this case,
a(1 2 g¯ ) 2 ( pA 2 eA ) 1 g¯ ( pB 2 eB )
qA 5 ]]]]]]]]]]
1 2 g¯ 2
a(1 2 g¯ ) 2 ( pB 2 eB ) 1 g¯ ( pA 2 eA )
qB 5 ]]]]]]]]]]
.
1 2 g¯ 2
Inequalities (3) and (5) are always satisfied. Inequalities (4) and (6) become
eA e (1 2 g¯ 2 )
eB < ]
1 ]]]
2 a(1 2 g¯ ) 1 pB 2 g¯ pA ; F(eA )
g¯
g¯
¯ A 2 e (1 2 g¯ 2 ) 1 g¯ (a(1 2 g¯ ) 2 pA 1 g¯ pB ) ; G(eA ).
eB > ge
Next, if g 5 0,
P. Belleflamme, F. Bloch / Int. J. Ind. Organ. 19 (2001) 1563 – 1582
1575
qA 5 a 2 pA 1 eA
q B 5 a 2 p B 1 eB .
Inequalities (4) and (6) are always satisfied while inequalities (3) and (5) become
2 eA e
eB < ]]
1]
2 a 1 pB ; H(eA )
g¯
g¯
¯ A 1 e 2 g¯ a 1 g¯ pA ; J(eA ).
eB < 2 ge
In Appendix A, we first use the two functions F(eA ) and G(eA ) to compute
prob( r 5 r0 ug 5 g¯ ), which measures the region of realizations of the shocks for
which the firm does not learn the value of g in the case where g 5 g¯ . We then
proceed in the same way for the case where g 5 0: functions H(eA ) and J(eA ) are
used to derive prob( r 5 r0 ug 5 0). Finally, we compute the ex ante probability that
the firm does not learn the truth as
W( pA , pB ) 5 r0 prob( r 5 r0 ug 5 g¯ ) 1 (1 2 r0 ) prob( r 5 r0 ug 5 0).
Notice that W( pA , pB ) is a quadratic form, which can be rewritten as
2
2
A( p A 1 p B ) 1 B( pA 1 pB ) 1 CpA pB 1 E
W( pA , pB ) 5 ]]]]]]]]]]]
,
8(1 2 g¯ 2 )e 2
with
A 5 2 g¯ [ g¯ (1 2 r ) 1 r (1 2 g¯ 4 ) ] , 0
B 5 2 g¯ (1 2 g¯ ) [ ( rg¯ 3 2 rg¯ 2 1 r 1 g¯ 2 2rg¯ )a 1 ( g¯ 1 r 2 rg¯ 3 2 rg¯ 2 )e ] . 0
C 5 2 g¯ 2 [ 2r (1 2 g¯ 2 ) 1 g¯ (1 2 r ) ] . 0
E 5 2(1 2 g¯ ) [ g¯ ( rg¯ 2 1 2rg¯ 2 rg¯ 3 2 g¯ 2 r )a 2 1 2 g¯ ( rg¯ 3 1 rg¯ 2 2 g¯ 2 r )ae
1 (4 1 4 g¯ 2 3rg¯ 3 2 g¯ 2 2 2rg¯ 2 2 rg¯ 2 rg¯ 4 )e 2 ].
It turns out that in the price-setting case, the firms’ information acquisition is
neither a monotonic function of prices, nor an increasing function of price
dispersion. As opposed to Alepuz and Urbano (1994) and Harrington (1995), firms
do not learn more by differentiating their prices. This difference stems from two
crucial differences in the models. First, we do not use the same parameter of
product differentiation as Alepuz and Urbano (1994) and Harrington (1995).
Second, we suppose that random shocks on inverse demand are independent,
introducing a correlation in the shocks affecting the demand function.
P. Belleflamme, F. Bloch / Int. J. Ind. Organ. 19 (2001) 1563 – 1582
1576
Step 3. Each firm’s expected profit in the first period is given by
EPB,i 5 ( pi 2 c)[a ( r0 ) 2 b ( r0 )pi 1 h( r0 )pj ]
1 d W( pA , pB )VB ( r0 ) 1 d [1 2 W( pA , pB )][ r0VB (1) 1 (1 2 r0 )VB (0)].
In order to obtain a Nash equilibrium, we impose the following condition on the
parameters: 3
8(1 2 g¯ 2 )e 2 [2b ( r0 ) 2 h( r0 )] 1 d (2A 1 C)[ r0VB (1) 1 (1 2 r0 )VB (0) 2VB ( r0 )] . 0
5
(15)
2
2
8(1 2 g¯ )e [a ( r0 ) 1 b ( r0 )c] 2 d B[ r0VB (1) 1 (1 2 r0 )VB (0) 2VB ( r0 )] . 0.
Notice that condition (15) is satisfied whenever the firms’ common discount
factor, d, is sufficiently low. As in Harrington (1995) and Mirman et al. (1994), a
pure strategy equilibrium exists when firms’ have a strong preference for the
present. Under condition (15), the first period game has a unique symmetric
equilibrium, with prices pA and pB equal to
2
2
8(1 2 g¯ )e [a ( r0 ) 1 b ( r0 )c] 2 d B[ r0VB (1) 1 (1 2 r0 )VB (0) 2VB ( r0 )]
p BE 5 ]]]]]]]]]]]]]]]]]
.
2
2
8(1 2 g¯ )e [2b ( r0 ) 2 h( r0 )] 1 d (2A 1 C)[ r0VB (1) 1 (1 2 r0 )VB (0) 2VB ( r0 )]
Recall that the equilibrium prices in a myopic Bertrand duopoly is given by
a ( r0 ) 1 b ( r0 )c
NE
p B 5 ]]]]].
2b ( r0 ) 2 h( r0 )
E
NE
It is easy to see that the sign of p B 2 p B is the same as the sign of
2 d [ r0VB (1) 1 (1 2 r0 )VB (0) 2VB ( r0 )] [B[2b ( r0 ) 2 h( r0 )]
1 (2A 1 C)[a ( r0 ) 1 b ( r0 )c]].
It is straightforward but tedious to show that
B[2b ( r0 ) 2 h( r0 )] 1 (2A 1 C)[a ( r0 ) 1 b ( r0 )c] . 0,
yielding the following Proposition.
Proposition 5. In a Bertrand duopoly, the firms raise their prices to reduce
information acquisition when the value of information is negative (i.e., if r0 ,
3
In our model, the profit function is not necessarily quasi-concave, and there is no guarantee that a
pure strategy equilibrium exists. The following conditions are sufficient conditions for the existence of
a Nash equilibrium in pure strategies.
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1577
f4(1 1 g¯ )(1 2 g¯ )2g /fg¯ (4 2 7g¯ 1 4g¯ 2 )g)
and lower their prices to increase information acquisition when the value of information is positive (i.e., if r0 . [4(1 1
g¯ )(1 2 g¯ )2 ] / [ g¯ (4 2 7 g¯ 1 4 g¯ 2 )]).
4.3. Comparison
We conclude this section by comparing the intensity of experimentation in the
price-setting monopoly and duopoly when information is valuable.
Proposition 6. If r0 .f4(1 1 g¯ )(1 2 g¯ )2g /fg¯ (4 2 7 g¯ 1 4 g¯ 2 )g and price is the
strategic variable, the intensity of experimentation is higher in a duopoly than in a
monopoly.
In our model, the comparison in Proposition 6 is trivially obtained because the
value of information for a price-setting monopolist is zero.
5. Prices vs. quantities
In this section, we compare experimentation in the Cournot and Bertrand
duopoly models. We focus on the case where information is valuable for Bertrand
duopolists (i.e., when condition (14) is met). In that case, the direction of
experimentation is identical in the two models: quantity-setting firms experiment
by increasing quantities (which leads to a decrease in prices) and price-setting
firms, by reducing prices (which leads to an increase in quantities). However,
given the complexity of the equilibrium solutions, the magnitude of experimentation cannot be directly compared in the two models. We thus provide a numerical
example to compare intensity of experimentation. In that example, we fix the
values of the parameters (a 5 1, c 5 0, r0 5 0.5, d 5 1, and e 5 10) and let the
degree of product differentiation, g¯ , vary from 0.77 to 0.9.4
Fig. 1 graphs the two measures of experimentation intensity, Mr and Mp , for the
Cournot and Bertrand duopolists.5
Fig. 1 suggests that the intensity of experimentation is higher in the Cournot
model. Both the deviation in the probability to learn the truth and the first-period
losses are higher for quantity-setting firms. These results are not easy to interpret.
They reflect differences in the value of information for Cournot and Bertrand
duopolists (note that for Bertrand duopolists, the value of information only become
positive for high degrees of product differentiation), and in the way strategic
4
One can easily check that all the assumptions needed to compute equilibria are satisfied in this
numerical example, and that condition (14) requires g¯ . 0.769 when r0 5 0.5.
5
As the strategic variables are not comparable, we do not measure deviations in prices or quantities.
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Fig. 1. Comparison of measures Mr and Mp for the Cournot and Bertrand duopolies.
variables determine the probability to learn the truth (the absolute probability to
learn the truth is much higher for Bertrand than for Cournot duopolists).
6. Conclusion
In this paper, we study learning and experimentation in a simple, unified model
of product differentiation. We compare the direction and magnitude of experimentation in four market settings: quantity-setting and price-setting monopoly,
Cournot and Bertrand duopoly. The quantity-setting monopoly and duopoly
always experiment by raising their quantities. It turns out that the intensity of
experimentation (as measured by the relative deviation from the myopic equilibrium of the probability to learn the truth and of the first-period quantities) is higher
in the quantity-setting monopoly than in the quantity-setting duopoly; furthermore,
P. Belleflamme, F. Bloch / Int. J. Ind. Organ. 19 (2001) 1563 – 1582
1579
the loss in first-period profits is smaller in the monopoly than in the duopoly. The
value of information to a price-setting monopoly is zero. In a price-setting
duopoly, the value of information may be positive or negative according to the
degree of product differentiation. When information is valuable, price-setting
duopolists experiment by lowering prices; when information is not valuable, they
raise their prices to limit information acquisition. Finally, as far as the comparison
between Cournot and Bertrand duopoly is concerned, a numerical example
suggests that the intensity of experimentation is higher in the Cournot model.
These comparisons show that the value of information to the firms depends both
on the market structure and the type of competition. This suggests that, in a
dynamic context, experimentation and learning might be affected by changes in
the market structure, like the entry of new firms or the launching of new products.
In turn, the market structure can be affected by the firms’ behavior and the
experimentation in quantities and prices. The construction and the study of a
dynamic model of interaction between learning and the market structure seem to
us to be an important topic for future research.
Acknowledgements
This paper was started when both authors were at HEC School of Management,
Jouy-en-Josas, France. We are grateful to Fondation HEC for financial support. We
have benefited from comments by participants at seminars in Warwick and at
ESEM 97 in Toulouse. We are grateful to the editor and two anonymous referees
for comments which greatly improved the paper.
Appendix A
In this appendix, we derive the ex ante probability that the firm does not learn
the truth in the price-setting case.
When g 5 g¯ , inequalities (4) and (6) respectively define the functions F(eA ) and
G(eA ), which are two increasing straight lines in the plane (eA , eB ). Given our
assumption e . [ g¯ (a 2 c)] /(1 2 g¯ ), these lines do not intersect for values of eA , eB
in [2e, e ] 2 . Hence, the region of realizations of the shocks for which the firm does
not learn the value of g can be represented by the hatched area in Fig. 2a. Since
the distribution of the noise is uniform, the area of the hatched region in Fig. 2a is
given by
ḡ
prob( r 5 r0 ug 5 g¯ ) 5 1 2 ]2 h[e (1 1 g¯ ) 1 a(1 2 g¯ ) 2 pA 1 g¯ pB ] 2
8e
1 [e (1 1 g¯ ) 1 a(1 2 g¯ ) 2 pB 1 g¯ pA ] 2 j.
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Fig. 2. Regions of realizations of the shocks for which the firm does not learn the true value of g. Panel
(a): case where g 5 g¯ ; Panel (b): case where g 5 0.
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When g 5 0, inequalities (3) and (5) respectively define the functions H(eA ) and
J(eA ), which are two decreasing straight lines in the plane (eA , eB ). Our assumption
e . [ g¯ (a 2 c)] /(1 2 g¯ ) guarantees that the two lines intersect for some value (eA ,
eB ) in [2e, e ] 2 , and that the two functions cut the axes as in Fig. 2b. The
probability that the firm does not learn the true value of g is given by the hatched
area in Fig. 2b,
ḡ
prob( r 5 r0 ug 5 0) 5 1 2 ]]]]
8e 2 (1 2 g¯ 2 )2
3 h2 g¯ [(1 2 g¯ )(e 1 a) 2 pb 1 g¯ pA ] [(1 2 g¯ )(e 1 a) 2 pA 1 g¯ pB ]
1 [(1 2 g¯ )(e 1 a) 2 pb 1 g¯ pA ] 2 1 [(1 2 g¯ )(e 1 a) 2 pA 1 g¯ pB ] 2 j.
Finally, we compute the ex ante probability that the firm does not learn the truth
as
W( pA , pB ) 5 r0 prob( r 5 r0 ug 5 g¯ ) 1 (1 2 r0 ) prob( r 5 r0 ug 5 0).
Using the previous expressions, we can rewrite W( pA , pB ) as the following
quadratic form
A( p 2A 1 p 2B ) 1 B( pA 1 pB ) 1 CpA pB 1 E
W( pA , pB ) 5 ]]]]]]]]]]]
,
8(1 2 g¯ 2 )e 2
with
A 5 2 g¯ [ g¯ (1 2 r ) 1 r (1 2 g¯ 4 ) ] , 0
B 5 2 g¯ (1 2 g¯ ) [ ( rg¯ 3 2 rg¯ 2 1 r 1 g¯ 2 2rg¯ )a 1 ( g¯ 1 r 2 rg¯ 3 2 rg¯ 2 )e ] . 0
C 5 2 g¯ 2 [ 2r (1 2 g¯ 2 ) 1 g¯ (1 2 r ) ] . 0
E 5 2(1 2 g¯ ) [ g¯ ( rg¯ 2 1 2rg¯ 2 rg¯ 3 2 g¯ 2 r )a 2 1 2 g¯ ( rg¯ 3 1 rg¯ 2 2 g¯ 2 r )ae
1 (4 1 4 g¯ 2 3rg¯ 3 2 g¯ 2 2 2rg¯ 2 2 rg¯ 2 rg¯ 4 )e 2 ].
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