Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol37.Issue2.Mar1999:

Duopoly experimentation: Cournot competition

* M. Dolores Alepuz, Amparo Urbano

`

Department of Economic Analysis, University of Valencia, Campus dels Tarongers, Avda. Dels Tarongers, Valencia, E-46022, Spain

Received 26 March 1996; received in revised form 28 February 1998; accepted 19 May 1998

Abstract

This paper analyzes learning behavior in an industry facing uncertainty. We consider a duopoly game where firms have imperfect information about market demand and they learn through observing market prices. The main body of our study consists of showing how firms make the price a more informative signal through their experimental behavior, and how this behavior compares to its monopoly counterpart. We extend previous analysis to the case where the demand unknown parameter takes values on the real line. We also find that experimentation under Cournot duopoly is smaller than under monopoly whenever the demand’s unknown parameter is sufficiently precise. 1999 Elsevier Science B.V. All rights reserved.

Keywords: Experimentation; Duopoly learning

JEL classification: C72; D83

1. Introduction

Firms acting under uncertainty may find it profitable to acquire relevant information. One way to acquire information about an uncertain parameter is via experimentation. Experimentation by a firm is the use of present actions to vary the amount of information available in the future. However, when these actions are observed by other firms they also affect their inferences about the same parameter and hence the underlying market competition. It is the purpose of this paper to study experimentation under competition. This is an extension of Mirman et al., 1993a who study monopoly experimentation, avoiding the competition effect.

We deal with duopoly thus introducing competition and we also consider a continuum of possible values for the slope of demand. Strategic interaction adds a new effect into

*Corresponding author. Tel.:134 6 3828246; fax:134 6 3828249; e-mail: amparo.urbano@uv.es 0165-4896 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved.

the analysis since the result of the experimenting behavior by one firm is observed by the rival. In such a setting and if information is valuable, is there any unilateral incentive to experiment? Does a monopoly experiment more or less than a duopoly?

To answer these questions we propose a two period horizon model played by two firms. They face a linear inverse demand with a random slope. Each period price is determined by the total quantities together with a random noise. The mean of the random slope is an unknown parameter and firms have the same subjective probability about it. They cannot precisely infer the true value of this parameter from the market price since it includes a random noise. The game proceeds as follows: in the first period firms choose simultaneously and independently production levels. These actions together with the noise term determine the market price. It is assumed that both the market price and the industry output are observable by the firms. In the second period, they again choose quantities based upon their updated beliefs about the unknown parameter. The expected payoffs of the firms are their two period profits.

By experimenting a firm increases the informational content of its market observa-tions, i.e. prices. However, in order to do that it must give up present period profits. In a duopoly setting firms affect the informativeness of the commonly observed signal. Thus

1

they may affect the degree to which a rival is likely to update. Since firms face a strategic informational choice, an important determinant of their behavior is whether expected future profits increase with an increase in information. It is well known that, in contrast to a single agent problem, an increase in information in a game may make some players worse off. We demonstrate that this cannot happen in models like ours, where the demand structure is linear and the a priori information is common.

The main contribution of this article is twofold. First, we generalize duopoly experimentation to the case of a continuum of possible demand curves. We prove that the informativeness of the commonly observed signal increases with a firm’s output. Second, we relate experimentation and market competition. We find out that what is relevant is the a priori uncertainty about the random demand. In particular, we show that if the random slope is sufficiently precise which, in turn, makes the commonly observed market signal precise as well, then the monopoly will experiment more than the Cournotian duopoly. The intuition behind this result can be understood by noting that, under duopoly, firms face a strategic informational choice in the sense of how informative to make the publicly observed market signal. If initial beliefs about the random slope are precise so is the market signal and then, posterior beliefs may became

1

Related to our work are the ‘signal-jamming’ models of Riordan, 1985; Fudenberg and Tirole, 1986; Mirman et al., 1993b. In these models no firm is perfectly informed about the state of nature and each firm may have an incentive to manipulate the inferences drawn by rival firms. The principal difference between these signal-jamming models and ours lies in the assumption made about the observability of actions: in signal-jamming, firms do not know – even ex-post – the action chosen by rivals. Therefore, firms attempt to influence the direction in which a rival updates its beliefs. Urbano, 1993 investigates a model with both experimentation and signal-jamming under duopoly.

2

too accurate by experimentation. Hence, the rival will be better informed and a tougher competitor: too precise signals discourage experimentation by duopolists.

The value of information in oligopoly games has been the subject of intensive research. These studies typically assume either that firms transmit information through ‘certifiable / verifiable announcements’ or that the signals that yield information to the

3

firms are exogenously generated. In contrast with them, our model endogenously determines the amount of information available to firms. The closest model to ours is the duopoly experimentation model of Mirman et al., 1994, who have also generalized the results of Mirman et al., 1993a. Their main contribution is to extend duopoly experimental behavior to mixed strategy equilibrium and to show some cases for which the net value of information for the duopoly is positive. However, they are unable to relate the duopoly experimental behavior with that of the monopoly in their general setting. Related to our model but with a different purpose are Aghion et al., 1993; Alepuz and Urbano, 1993; Harrington, 1995. They assume that the degree of substitution between products is unknown to explain the phenomenon of price dispersion under oligopoly. Finally, Alepuz and Urbano, 1997 deal with experimental behavior but in asymmetric heterogeneous duopoly markets.

The plan of the paper is as follows. The model and informational assumptions are laid down in Section 2. Then, we analyze the main features of information and the experimental behavior of firms. Section 3 is devoted to show that larger outputs lead to more informative market signals. Monopoly and duopoly experimentation are compared in Section 4. Some concluding remarks are given at the end.

2. The Cournot model

Consider a two period duopoly model. The firms produce a homogeneous product over the two periods. Inverse market demand in each period is given by

1 2

˜ ˜ _˜

P 5_a2u_{( q} 1_{q )}1´

t t t t

i

˜

where P is the price in period t, t5_{1, 2 and q , i}5_{1,2, is firm i’s quantity in period t.}

t t

˜ _˜

The parameter u _{is the fixed random slope and} ´ _{is each period’s random demand}

t

shock. It is assumed that each of these two random variables has full support on R and that they are independently and normally distributed. These distributions are known by

2

Note that here, in contrast to the duopoly models of information sharing, the key question is not whether firms are willing or not to share information but how much information to produce, since all the information contained in the market signal is immediately shared with the rivals. See Malueg and Tsutsui, 1996 and references herein.

3

See Clarke, 1983; Gal-Or, 1985, 1986; Fried, 1984; Li, 1985; Logan, 1988; Novshek and Sonnenschein, 1982; Shapiro, 1986; Vives, 1984 among others.

the firms. Firms’ market interaction is modeled as a game of imperfect information. The game unfolds as follows:

Description of the game. The game consists of two periods. In period one we

distinguish two stages:

In stage 1, Nature chooses valuesu _and´ _{of the two independent random variables}

1

2

˜ _˜ ˜

u|_{N(m, h) and} ´ |_N(0, t_{) respectively, where h}5_{1 /(}s _{) is} u_{’s precision and}

1 u

2

˜

t 5_{1 /(}s _{) is} ´_{’s precision. As we noted above the Normal distributions of these two}

´ 1

random variables are common knowledge, but not their realizations.

1 1

In stage 2, firms choose quantities q , q simultaneously and independently. These1 2 1 2

quantities become commonly known. Let Q 5_q 1_{q , then the price P is realized}

1 1 1 1

where

P 5_a2u_Q 1´ ₍₁₎

1 1 1

Firms observe P but notu _and´_.

1 1

Period two consists of two similar stages:

˜

In stage1, Nature selects a value ´ _{of the random variable}´_{, which has the same}

2 2

1 2

˜

distribution as that of ´_{. This selection is independent of P , q and q .}

1 1 1 1

1 2

In stage 2, firms choose quantities q , q simultaneously and independently (which2 2 1 1

are functions of P and Q ). Let Q 5_q 1_{q . Then the price P is determined by}

1 1 1 2 2 2

P 5_a2u_Q 1´ ₍₂₎

2 2 2

which, as before, is observed by the firms.

We assume that production cost are zero (or that P is the net price received in t, aftert

subtracting a constant marginal cost that is the same for both firms), and that firms seek to maximize the sum of the two period expected profits.

Our analysis is confined to the pure strategy subgame perfect equilibria of this two period game. The equilibrium is calculated in the backward induction way. We begin by

1 2

analyzing the equilibrium of the second period as a function of P , q and q . After1 1 1

1 2 1 2

ˆ ˆ

observing P , q and q , both players choose q , q1 1 1 2 2 such that:

i i 1 2

ˆq 5_{Argmax q E[P /P , q , q ] i}5_{1, 2 (3)}

2 2 2 1 1 1

1 2

To solve this problem we have to find out first what is E[P /P , q , q ] i.e.2 1 1 1 1 2

˜ _˜

E[a2u_Q 1´_{/P , q , q ]. Since}

2 1 1 1

1 2 1 2

˜ _˜ ˜

E[a2u_Q 1´_{/P , q , q ]}5_a2_E[u_{/P , q , q ] (4)}

2 1 1 1 1 1 1

1 2

˜

it is sufficient to find E[u_{/P , q , q ]. By Eq. (1)}

1 1 1

˜ a2_P ´

1 _˜

]]5u 2] ₍₅₎

Q1 Q1

Therefore for each value of the parameteru_,

˜ a2_P

1 2

]]|_N(u_,t_{Q ) (6)}

1

˜

The value ofu _{is unknown to firms, but they a priori believe that} u|_{N(m,h). Then,}

after observing P and knowing Q , their new belief aboutu _{is that it is a realization of a}

1 1

ˆ ˆ

random variable which is Normally distributed according to N(u_{, h ) (see DeGroot, 1970,}

4

page 167), where

a2_P

1

2 _]]

mh1t_Q

S

D

1 _{Q mh1tQ (a2P )}

1 1 1

ˆ ]]]]]] ]]]]]

u 5 5 ₍₇₎

2 2

h1t_{Q h}1t_Q

1 1

and

2

ˆ

h5_h1t_{Q (8)}

1

1 2 1 2

˜ ˜ ˜

Hence, E[u_{/P , q , q ]}5u _{and by Eq. (4), E[P /P , q , q ]}5_a2u_{Q . Inserting this}

1 1 1 2 1 1 1 2

result in Eq. (3), we obtain the following unique symmetric solution:

a

1 2 _ˆ

ˆ ˆ ]

q 5_q 5_q(u₎5 ₍₉₎

2 2 _ˆ

3u

Hence, both firms equilibrium expected profits in the second period are functions

ˆ

V(u_{), where}

2

a

ˆ ˆ ˆ ˆ ]

V(u₎5_q(u_{) a}

f

2₂u_q(u₎

g

5 ₍₁₀₎

ˆ

9u

Therefore,

ˆ Lemma 1. V(.) is a decreasing, strictly convex function of u_.

The intuition of this lemma is clear. The first property of V(.) is that the higher the

ˆ

value of u_{, the mean of the demand’s unknown slope, the smaller the second period}

˜

expected profits to each firm. The convexity of V(.) means that information about u _is

˜

valuable for each duopolist from an ex-ante viewpoint. Letu 5u _{and let V(}u_{) be each}

5

˜ ˜

duopolist’s second period profits whenu 5u _{is known. To see that information about}u

is valuable for firm i, it suffices to compare between firm i’s expectations over second

˜

period profits when the true value ofu _{is to be learnt between periods one and two and}

when it is not. In the former (‘informed’) case second period expected profits are equal to:

4 _ˆ

Note that given our normality assumptions, the signal and updated beliefs,u, may take on negative values. Firms are constrained to choose positive prices and quantities. For convenience, we ignore this and, given the firms’ strategies that we derive, we can get negative prices and outputs for certain combinations of the signal

ˆ

andu. The probability of such an event can be made arbitrarily small by appropriately choosing the variances of the model.

5 2 i

V(u)5q(u)[a22uq(u)]5(a ) /(9u), where q (2 u) is firm i’s profits maximizer quantity in period two and

1 2

1`

˜

E

f

V(u₎

g

5

E

_V(u_)g(u_{) d}u

2`

˜ ˆ ˆ

(where g is the density function ofu_{which is Normally distributed according to N(}u_{, h ))}

and in the latter (‘uninformed’) case second period expected profits are simply

˜ ˆ

V [E(u_)]5_V(u_).

˜

Definition 1. Information about u _{is valuable for firm i, i}5_{1, 2, if}

˜ ˜ ˆ

E[V(u_)]._{V [E(}u_)]5_V(u_{) (11)}

6 _ˆ

The strict convexity of V(.) in u _{guarantees, by Jensen’s Inequality, that Eq. (11) is}

satisfied.

2.1. The analysis of the first period

Given the analysis of the second period, one can write each firm’s two period expected profits as a function of first period outputs. Denote first

1 2 i

˜ _˜

E[(a2u_{( q} 1_{q )}1´_{)q ] i}5_{1, 2 if P} .₀

i 1 2 _{1 1 1 1}

p _{( q , q )}5

H

₍₁₂₎

1 1 1

0 otherwise to be the first period expected profits and let

i 1 2 i 1 2 _ˆ

P _{( q , q )}5p _{( q , q )}1_E[V(u_{)] (13)}

1 1 i 1 1

be the two period profits to firm i. Observe that

1`

ˆ ˆ

E[V(u_)]5

E

_V(u_{(P , Q ))f(a}2_{P ) dP (14)}

1 1 1 1

2`

˜

where f is the density function of the random variable (a2_{P ). Since}

1

1 2

˜ ˜ _˜ ˜ _˜

(a2_{P )}5u_{( q} 1_{q )}1´ 5 u_Q 1´_.

1 1 1 1

then, for each value of Q ,1

ht

˜ ]]]

(a2_{P )}|_{N mQ , . (15)}

1

S

1 2

D

h1t_Q

1

1 2

ˆ ˆ

The problem of the firms is to choose q1 and q , such that1

6

An alternative explanation of the relationship between the convexity of V(.) and the value of information is the following: V(.) convex means that firms are risk-loving so that they prefer more variability in their posterior

˜

mean on u. This can be achieved by increasing the amount of new information concerning the slope of demand. See Harrington, 1995.

1 1 2 2 1 2

ˆ ˆ ˆ ˆ

q [ArgmaxP_{( q , q ) , q [Argmax}P_{(q , q ) ,}

1 1 1 1 1 1 1 1

(16)

1 2

q1 q1

1 2

¯ ¯

Let q1 and q1 be the first period myopic solutions. Namely,

1 1 1 2 2 2 1 2

] ] ] ]

q [Argmaxp _{( q , q ) , q [Argmax}p _{(q , q ) ,}

1 1 1 1 1 1

(17)

1 2

q1 q1

1 2 1 2

ˆ ˆ ¯ ¯

We are interested in comparing (q , q ), with (q , q ). Suppose that the conditions1 1 1 1 7

for the existence of first period subgame perfect equilibria in pure strategies are fulfilled. Next, we characterize any selection of these equilibria.

1 2

¯ ¯

The first order conditions for the myopic solutions q , q1 1 set up current marginal

1 2

ˆ ˆ

profits equal to zero, meanwhile the first order conditions for q , q , are:1 1

i i j _ˆ

ˆ

≠p _{( q , q )} ≠_E[V(u_)]

1 1 1

]]]]1]]]5_{0 i}5_{1, 2, i}±_{j (18)}

i i

≠_q ≠_q

1 1

i

ˆ

The first result of this paper is to show that E[V(u_{)] is an increasing function of q for}

1

i5_{1, 2, which implies that each firm increases output with respect to the myopic choice}

for experimental purposes. Thus, lemma 2 describes the effect of the incentives to manipulate information on the best response maps of the firms.

Lemma 2.

ˆ

≠_E[V(u_)] ]]] ._{0 .}

i

≠_q

1

Proof. By Eq. (14):

ˆ ˆ ≠_f(a2_{P )}

≠_E[V(u_)] ≠u

1

ˆ ˆ

]]]5

E

F

_V9₍u₎] _f(a2_{P )}1_V(u₎]]]

G

_{dP (19)}

i i 1 i 1

≠_q ≠_q ≠_q

1 1 1

In Appendix A we show that Eq. (19) can be expressed in the following way:

ˆ ˆ

≠_E[V(u_{)] 1} ≠u₎

ˆ

]]]5

E

]_V0₍u₎

F

2]

G

_f(a2_{P ) dP (20)}

i _{ˆ ≠P} 1 1

≠_{q1 h} 1

3 2

ˆ ˆ ˆ

By lemma 1, V(u_{) is strictly convex. In fact V}0₍u₎5_{(2a ) / 9}u _{. By Eq. (7),}

ˆ t_Q

≠u

1

]5 2]]],_{0 ,}

2

≠_{P h1tQ}

1 ₁

thus,

7 i

2

ˆ _2a t_Q ≠_E[V(u_{)] 1}

1

]]]5]]

E

] _f(a2_{P ) dP} ._{0 (21)}

i _ˆ2 _ˆ3 1 1

≠_{q 9h} u

1

i j

where Q 5_q 1_{q . j}

1 1 1

Lemma 2 comports well with previous results in the literature. In particular, it is a generalization of the two values case in Mirman et al., 1993a to a duopoly setting. It is also an alternative and a more general approach to Grossman, Kilhstrom and Mirman’s

8

paper who dealt with the n values’ case and with a normally distributed shock. A result similar to ours is obtained by Mirman et al., 1994 who also extended the model of Mirman et al., 1993a to a duopoly market. However, since their analysis is carried out under the assumption of two values of the unknown parameter, our model can be considered as well as a generalization of their results to the continuum case. The next proposition extends the results from best-replay mappings to equilibrium quantities. In particular, it shows that at the equilibrium the Cournotian duopolists experiment by increasing quantity with respect to their myopic choice.

1 2 1 1

¯ ¯ ˆ ¯

Proposition 1. Let (q , q ) be the myopic Cournot quantities. Then q ._{q and}

1 1 1 1

2 2 i

ˆ ¯ ˆ

q ._{q . In particular, q} .₀

1 1 1

Proof. By Eq. (18)

ˆ a 1 1 ≠_E[V(u_)]

i j

ˆ ] ]_ˆ ] ]]]

q 5 2 _q 1 _i5_{1, 2, i}±_{j (22)}

1 _{2m 2} 1 _{2m ≠} i

q1 1 2

ˆ

E[V(u_{)] depends on Q} 5_q 1_{q , hence}

1 1 1

ˆ ˆ

≠_E[V(u_)] ≠_E[V(u_)] ]]]5]]]_.

1 2

≠_q ≠_q

1 1

1 2

ˆ ˆ

Then by Eq. (22) q 5_{q , and by Eqs. (20) and (21),}

1 1

2 _ˆ

2a t_Q

a 1 1

1 2

ˆ ˆ ] ]] ]

q 5_q 5 1

E

_f(a2_{P ) dP (23)}

1 1 _{3m ˆ}2 _ˆ3 1 1

27mh u

1 2

ˆ _{ˆ ˆ}

where Q 5_q 1_{q .}

1 1 1

On the other hand, by solving Eq. (17) one shows that,

a

1 2

] ] _]

q 5_q 5 ₍₂₄₎

1 1 _3m

1 1 2 2

ˆ ¯ ˆ ¯

Consequently by Eq. (23); q ._{q , q} ._{q , and}

1 1 1 1

8

Grossman et al., 1977 (GKM) examine a consumer whose utility is given by z_] 1bx1´where´is random, z and x are quantities of consumption goods, and b[hb,b_] j. They show that the consumer increases consumption of x to spread the mean utility functions further apart and hence increase the informativeness of realized utility.

2 _ˆ

4at_Q

2a 1 1 2a ]

ˆ ] ]] ] ]

Q 5 1

E

_f(a2_{P ) dP} . 5_{Q j}

1 _{3m ˆ}2 _ˆ3 1 1 _3m 1

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 ofu_{. 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}2_{P ) /(Q ). Let}

1 1

S 5_(a2_{P ) /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 be1 2 1 2

the corresponding signals – random variables – of the parameteru_{, generated by these}

outputs.

Observe that by Eq. (5), for any value x of the total output,

2

ht_x ]]

SxN m,

S

2

D

(25)

h1t_x

ˆ

Moreover, by Eq. (7), these signals Sx and Sx lead to some posterior aboutu_,u_{(Sx )}

1 2 1

ˆ

andu_{(Sx ), respectively.}

2

2 2

ˆ ˆ ˆ ˆ

Defineu_{(x, s)}5_(mh1t_{x s) /(h}1t_{x ) and notice that}u 5u_{(Q , S ) for}u_{as in Eq. (7).}

1 Q₁

The next lemma shows that the precision of a signal and hence its informative content increases in x.

ˆ ˆ

E

f

F(u_{(x , Sx ))}

g

._E

f

_F(u_{(x , Sx ))}

g

₍₂₆₎

Sx₁ 1 1 Sx₂ 2 2

ˆ ˆ

for any strictly convex function F(u_{), i.e. for any strictly convex function F(}u_),

ˆ ˆ

E

F(u_{(x , s))f(x , s) ds} .

E

_F(u_{(x , s))f(x , 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 output1 1 _]

]

respectively. Then, the signal S5_{Sˆ is more informative than the signal S}5_{S .}

Q₁ Q₁

] ]

ˆ ˆ ˆ ] ˆ

Proof. By lemma 3 it suffices to prove that E F(_ˆ

f

u_{(Q , S ))}

g

._{E F(}

f

u_{(Q , S )) for any}

g

S 1 S 1

strictly convex function F. _{] ]}

ˆ ] ˆ

Consider the signal S5_{Sˆ and S}5_{S . By proposition 1, Q} ._{Q and since by}

Q₁ Q₁ 1 1

ˆ

Proposition 2, E F(_ˆ

f

u_{(x, Sx))}

g

_{is an increasing function of x, whenever F is strictly}

S _{] ]}

ˆ ˆ ˆ ] ˆ

convex, then E F(_ˆ

f

u_{(Q , S ))}

g

._{E F(}

f

u_{(Q , S )) for any strictly convex function F.}

g

_j

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 will1

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. Ifu _{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 beliefsu, are also quite precise. In

2 2

˜

particular, by Eq. (25) the variance of the market signal ((a2P )) /(Q ) is ((1 tQ11h) /(htQ )), and by Eq. (7)1 2

2 4 _˜

ˆ ˆ

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[V(u_{(Q ))] 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[V(u_{(Q ))] 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 Q1 and Q1 be the monopoly solution for the first period and the myopic

M _ˆ

output respectively. Denote by V (u_{) the monopoly expected profits for the second}

M _ˆ 2 _ˆ M _ˆ

period. It is easily verified that V (u₎5_{(a ) /(4}u_{) and hence V (}u_{) is strictly convex.}

M _ˆ

For any first period monopoly output x, let Sx5_(a2_{P ) /x, and let E [V (}u_{(x, 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 (u_{(x, Sx))] is increasing in x. In Appendix A it is shown:}

S_x

M M

ˆ ¯

Lemma 4. Q ._Q 5_{a /(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

i 1 2 D _{ˆ ˆ}

chooses q , i5_{1,2, and where Q} 5_{( q} 1_{q ), and let V (}u_{(Q ))}5_2V(u_{(Q )), 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[V(}u_{(Q ))] is concave in Q.}

1 1 1 1

M i

ˆ

Proof. First, let us show that Q 1_{a /(6m)}._{q , i}5_{1, 2. Indeed, since the first period}

1 1

ˆ

demand is P5_a2u_Q1´_{, then E[P]}5_a2_{mQ and hence a}2_mQ $_{0. Therefore, Q} #_{a /}

1 1

M M

M M 1

¯ ¯ ˆ ] ˆ ˆ

m5_{2Q . By lemma 3, Q} #_{Q . Thus, Q} #_{Q . This implies that}

1 1 1 2 1 1

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 1_{a /(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 Q1 1 is optimal for the monopoly,

a q

M M M

M _ˆ M _{ˆ ˆ} M _{ˆ ]} M _{ˆ ˆ ]}

p _{(Q )}1_E

f

_{V (}u_{(Q ))}

g

$p

S

_Q 2

D

1_E

F

_V

S

u

S

_Q 2

DDG

1 1 1 1 1 _6m 1 _6m

(28)

i _ˆ i 2

ˆ ˆ ˆ

and since q 5_{1 / 2Q and (q , q ) is the first period Nash equilibrium output,}

1 1 1 1

a

M

1 1 2 _{ˆ ˆ} D _ˆ 2 2

ˆ ˆ ] _{ˆ ˆ}

p _{(q , q )}1_{E V(}

f

u_{(Q ))}

g

$p

S

_Q 1 2_{q , q}

D

1 1 1 1 1 1 _6m 1 1

q

M

M _{ˆ ˆ ]}

1_E

F

_V

S

u

S

_Q 2

DDG

₍₂₉₎

1 _6m

a

M

2 1 2 _{ˆ ˆ} D _ˆ 1 1

ˆ ˆ ] _{ˆ ˆ}

p _{(q , q )}1_{E V(}

f

u_{(Q ))}

g

$p

S

_Q 1 2_{q , q}

D

1 1 1 1 2 1 _6m 1 1

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 _ˆ D _{ˆ ˆ} D _{ˆ ] ˆ} D _{ˆ ˆ ]}

p _{(Q )}1_{E V (}

f

u_{(Q ))}

g

$p

S

_Q 1 _,Q

D

1_E

F

_V

S

u

S

_Q 2

DDG

1 1 1 1 _6m 1 1 _6m

(31) Eqs. (28) and (31) can be rewritten as follows

a a

M M

M _ˆ M _{ˆ ]} M _{ˆ ˆ ]} M _{ˆ ˆ}

p _{(Q )}2p

S

_Q 2

D

$_E

F

_V

S

u

S

_Q 2

DDG

2_E

f

_{V (}u_{(Q ))}

g

₍₃₂₎

1 1 1 1 _6m 1 _6m 1

11

Notice that under this situation a first period duopoly equilibrium in pure strategies will always exist, i.e. Eq.

i

a a

M M

D _{ˆ ˆ} D _{ˆ ˆ ]} D _{ˆ ] ˆ} D _ˆ

E V (

f

u_{(Q ))}

g

2_E

F

_V

S

u

S

_Q 1

DDG

$p

S

_Q 1 _{, Q}

D

2p _{(Q )}

1 1 _6m 1 1 _6m 1 1 1

(33) To derive a contradiction it is sufficient to show that, by Eqs. (32) and (33):

a M M

M _{ˆ ˆ ]} M _{ˆ ˆ} D _{ˆ ˆ} D _{ˆ ˆ}

E

F

V

S

u

S

_Q 2

DDG

2_E

f

_{V (}u_{(Q ))}

g

._{E V (}

f

u_{(Q ))}

g

2_E

F

_V

S

u

S

_Q

1 _6m 1 1 1

a

]

1

DDG

_(A)

6m and

a a

M M

M _ˆ M _{ˆ ]} D _{ˆ ] ˆ} D _ˆ

p _{(Q )}2p

S

_Q 2

D

#p

S

_Q 1 _{, Q}

D

2p _{(Q ) (B)}

1 1 1 1 _6m 1 1 _6m 1 1 1

Let us first prove (A).

M _ˆ

Notice that if E[V (u_{(Q ))] is concave in Q , then}

1 1

a M

M _{ˆ ˆ ]} M _{ˆ ˆ} M _{ˆ ˆ}

E

F

V

S

u

S

_Q 2

DDG

2_E

f

_{V (}u_{(Q ))}

g

$_{E V (}

f

u_{(Q ))}

g

1 _6m 1 1

a

M

M _{ˆ ˆ ]}

2_E

F

_V

S

u

S

_Q 1

DDG

₍₃₄₎

1 _6m

M M

ˆ

Next, let P 5_a2u_{Q 1}1´_{, P} 5_a2u_(Q 1_{a /(6m))}1´ _{and let}

1 1 1

ˆ

mh1t_{Q (a}2_{P )}

1 1

ˆ ]]]]] u 5

2

ˆ h1t_Q

1

and

a

M M

ˆ ]

mh1t

S

_Q 1

D

_(a2_{P )}

1 1

M _6m

ˆ ]]]]]]]]]

u 5 _.

2

a

M

ˆ ]

h1t

S

_Q 1

D

1 _6m

M _ˆ 2 _ˆ D _ˆ 2 _ˆ

Also recall that, V (u₎5_{(a ) /(4}u_{), V} 5_2V(u₎5_{(2a ) /(9}u_{), then define:}

a

M M

M M _{ˆ ˆ} D _{ˆ ˆ ]}

W 5_{E V (}

f

u_{(Q ))}

g

2_E

F

_V

S

u

S

_Q 1

DDG

5

1 1 _6m

M

M _ˆ M _ˆ M M

E

V (u_)f(a2_{P ) dP} 2

E

_{V (}u _)f(a2_{P ) dP} 5

1 1 1 1

2

a 1 1 M M

]

F

E

] _f(a2_{P ) dP} 2

E

] _f(a2_{P ) dP}

G

.₀

1 1 M 1 1

4 uˆ uˆ

M

ˆ ˆ

by the proof of lemma 3, since Q ._Q 1_{a /(6m). Also, define:}

1 1

a

M M

D D _{ˆ ˆ} D _{ˆ ˆ ]}

W 5_{E V (}

f

u_{(Q ))}

g

2_E

F

_V

S

u

S

_Q 1

DDG

5

M

D _ˆ D _ˆ M M

E

V (u_)f(a2_{P ) dP} 2

E

_{V (}u _)f(a2_{P ) dP} 5

1 1 1 1

2

2a 1 1 M M

]

F

E

] _f(a2_{P ) dP} 2

E

] _f(a2_{P ) dP}

G

.₀

1 1 M 1 1

9 uˆ uˆ

again by lemma 3. Hence

2

1 1 a 2a2

M D _{] ]} M M

F

_{] ]]}

G

W 2_W 5

F

E

_f(a2_{P ) dP} 2

E

_f(a2_{P ) dP}

G

2

1 1 M 1 1

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 P5_a1_{(mh) /}t_{Q be the price which makes posterior beliefs} u 5_{0, then the}

1

ˆ

probability ofu ,_{0 is arbitrarily small whenever it is small the probability of any price}

˜ ˜

¯ ¯

observation P._{P. In particular, notice that P}5_{E[P ]}1_{var(P )(mh) /(Q ) and hence}

1

12

˜ ˜

¯

Prob(P._{P )}5_Prob(P2_{E[P ]}._{var(P )(mh) /Q ) which by Chebyshev’s Inequality, by}

1

˜ ˜

the definition of var(P ) and for var(P )(mh) /(Q )._{0 has the following upper bound:}

1

2

t_Q

mh 1

ˆ ] ]]

Prob P

S

2_{E[P ]}.var(P)

D

, . 2

Q1 m hˆ

This upper bound decreases with h and m, i.e. with a more precise (and with a higher

˜

mean) random slopeu_{. 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 ((a2_{P )) /(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 P*5_a1_{(mh) /(}t_{Q )((3h}2_{2h ) /(3h}1_{h )) be the price at which} u 5_{(3mh) /(3h}1

1

˜

ˆ ˆ ˆ ¯

h )._{0 (or (h}1_{(h ) / 3)}u 5_{mh), and with P*},_{P. The next lemma shows that if} u _is

ˆ

precise enough, then E[V(u_{(Q ))] is concave in Q.}

ˆ

Lemma 5. A sufficient condition for E[V(u_{(Q ))] to be concave in Q is that the variance}

12

Chebyshev’s Inequality: Let X be a random variable, andd .0. Then Var(X )

]]

Prob(X2E[X].d),

2

13 _˜ d _˜ 2 2 _ˆ

˜

Recall that P5a2uQ1´, and then var(P )5(Q ) /(h)11 /t 5(tQ ) /(ht)5(h ) /(ht) where h andt are the

˜ _˜ ˜

precisions ofuand´respectively. Note that var(P ) is a decreasing function of both h andt. Also, recall that,

2 2

Hence by Eq. (A.12), Eq. (A.11) is

ˆ

ˆ

≠

_E[V(

u

_)]

₁

≠u

ˆ

]]]

5 2

]

_.

E

_V

0

₍

u

₎

]

_f(a

2

_{P ) dP}

i

_h

≠

_P

1 1

≠

_q

₁

1

2

ˆ

Q

≠u

1

ˆ

]

]

2

E

_V

0

₍

u

₎

F G

_f(a

2

_{P ) dP}

5

1 1

h

≠

_P

1

ˆ

ˆ

1

≠u

≠u

ˆ

]

]

]

2

E

_V

0

₍

u

₎

F

₁

1

_Q

G

_f(a

2

_{P ) dP}

1 1 1

h

≠

_P

≠

_P

1 1

That by Eq. (A.5) yields:

ˆ

ˆ

≠

_E[V(

u

_)]

₁

≠u

ˆ

]]]

5

E

_V

0

₍

u

₎

]

F

2

]

G

_f(a

2

_{P ) dP}

i

_ˆ

_≠

_P

1 1

≠

_q

₁

_h

1

2

ˆ

where h

5

_h

1t

_{Q .}

1

Proof of Lemma 3. First observe that E[Sx ]

5

_{E[Sx ]}

5

_{m (see Eq. (25)). Hence it is left}

1 2

to prove that if Eq. (26) holds then the precision of Sx is higher than that of Sx .

1 2

Suppose that:

1` 1`

ˆ

ˆ

E

F(

u

_{(x , s))f(x , s) ds}

.

E

_F(

u

_{(x , s))f(x , s) ds}

_(A.13)

1 1 2 2

2` 2`

for any strictly convex function F.

ˆ

ˆ

Observe that by the definition of Sx and

u

_{, the function}

_e

_F(

u

_{(x, s))f(x, s)ds can be}

expressed as,

1` 1`

ˆ

ˆ

E

F(

u

_{(x, s))f(x, s) ds}

.

E

_F(

u

_{(x, a}

2

_{P ))f(x, a}

2

_{P ) dP}

_(A.14)

1 1 1

2` 2`

Let next show that the righthand side of Eq. (A.14) is an increasing function of x. First

notice that Eq. (20) holds if we replace V by F. Secondly (by Eq. 7, in the main text),

ˆ

(

≠u

_{(x, a}

2

_{P ) /}

≠

_P

,

_{0. Hence by the strict convexity of F,}

1 1

ˆ

≠

₁

≠u

ˆ

ˆ

]

F

E

_F(

u

_{(x, a}

2

_{P ))f(x, a}

2

_{P ) dP}

G

5

]

E

_F

0

₍

u

₎

F

2

]

G

_{f(x, a}

1 1 1

≠

_x

_h

_ˆ

≠

_P

1

2

_{P ) dP}

.

₀

_(A.15)

1 1

ˆ

ˆ

where

u 5 u

_{(x, a}

2

_{P ).}

1

ˆ

By Eqs. (A.14) and (A.15),

e

F(

u

_{(x, s))f(x, s) dsis an increasing function of x, for any}

strictly convex function F. Hence by Eq. (A.13) it must by that x

.

_{x .}

1 2

2 2 2

h

t

_x

≠

_h

t

_x

_{2h x}

t

]]

] ]]

]]]

Sx

|

_{N m,}

S

D

_and

S

D

5

.

_{0 .}

2

_≠

_x

2 2 2

h

1

t

_x

_h

1

t

_x

_(h

1

t

_{x )}

Thus, the precision of Sx increases in x.

Proof of Lemma 4.

M

Let

p

_{(Q )}

5

_(a

2

_{mQ )Q , be the monopoly first period expected profits. Then}

1 1 1 1

a

]

_{M M}

]

Q

5

_Argmax

p

_{(Q )}

5

_(A.16)

1 1 1

_2m

Q₁

M _M

ˆ

ˆ

Q

5

_{Argmax [}

p

_{(Q )}

1

_E[V(

u

_{(Q , a}

2

_{P ))]]}

_(A.17)

1 1 1 1 1

Q₁

The first order condition for Eq. (A.17) is,

M

_ˆ

≠p

≠

_E[V(

u

_{(Q , a}

2

_{P ))]}

1 1 1

]]

1

]]]]]]

5

_{0 .}

≠

_Q

≠

_Q

1 1

ˆ

Since E[V(

u

_{(Q , a}

2

_{P ))] is increasing in Q , (see the proof of lemma 3) and since}

1 1 1

M

M M

_ˆ

_¯

M

(

≠p

_{) /(}

≠

_{Q )}

,

_{0 (}

p

_{is strictly concave in Q ), then Q}

.

_Q

5

_{(a) /(2m).}

_j

1 1 1 1 1 1

Proof of (B) of Proposition 4.

We have to prove:

a

a

M M

M

_ˆ

M

_ˆ

_]

D

_ˆ

_]

_ˆ

D

_ˆ

p

_{(Q )}

2

p

S

_Q

2

D

#

p

S

_Q

1

_{, Q}

D

2

p

_{(Q )}

1 1 1 1

_6m

1 1

_6m

1 1 1

a

M

M M

_ˆ

M

_ˆ

_]

Dp

5

p

_{(Q )}

2

p

S

_Q

2

D

1 1 1 1

_6m

a

a

M M

ˆ

ˆ

ˆ

]

ˆ

]

5

_(a

2

_{mQ )Q}

2

S

_a

2

_m

S

_Q

2

DDS

_Q

2

_m

D

5

1 1 1

_6m

1

₆

a

a

a

M M

ˆ

ˆ

ˆ

ˆ

ˆ

]

]

ˆ

]

(a

2

_{mQ )Q}

2

_(a

2

_{mQ )Q}

1

_(a

2

_{mQ )}

2

S

_Q

2

D

1 1 1 1 1

_6m

₆

1

_6m

and

a

M

D D

_ˆ

_]

D

_ˆ

Dp

5

p

S

_Q

1

D

2

p

_{(Q )}

1 1

_6m

1 1

a

a

M M

ˆ

]

ˆ

ˆ

]

ˆ

ˆ

5

S

_a

2

_m

S

_Q

2

DDS

_2Q

2

_Q

1

D

2

_(a

2

_{mQ )Q}

1

_6m

1 1

_3m

1 1

Hence:

a

a

a

M M

a

M D

_ˆ

_]

_]

_ˆ

_]

_ˆ

_ˆ

_ˆ

_]

Dp

2

Dp

_(a

2

_{mQ )}

2

S

_Q

2

D

2

s

_a

2

_mQ

d

S

_Q

2

_Q

1

D

a

M

a

ˆ

ˆ

]

]

1

S

_2Q

2

_Q

1

D

5

1 1

6

3m

a

a

M

a

M M

a

ˆ

]

]

ˆ

ˆ

]

ˆ

ˆ

ˆ

]

(a

2

_{mQ )}

2

S

_Q

2

_Q

2

D

2

s

_a

2

_mQ

d

S

_Q

2

_Q

1

D

1

_6m

₆

1 1

_6m

1 1 1

_3m

a

M

a

ˆ

ˆ

]

]

1

S

_Q

2

_Q

1

D

5

1 1

6

3m

a

M

a

M

a

ˆ

ˆ

ˆ

ˆ

]

s

_a

2

_mQ

2

_a

1

_mQ

d

2

]

S

_Q

2

_Q

2

]

D

1 1 1 1

6m

6

6m

a

a

a

M M M

ˆ

ˆ

ˆ

]

]

ˆ

ˆ

]

2

s

_a

2

_mQ

d

S

_Q

2

_Q

1

D

1

S

_Q

2

_Q

1

D

5

1 1 1

_6m

₆

1 1

_3m

2

a

M

a

M

a

a

M

a

M

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

]

s

_Q

2

_Q

d

2

]

S

_Q

2

_Q

2

]

D

1

]]

1

S

_Q

2

_Q

2

]

D

s

_a

2

_mQ

d

5

1 1 1 1 1 1 1

6

3

6m

36m

6m

a

a

M M

ˆ

ˆ

]

ˆ

]

S

Q

2

_Q

2

DS

_a

2

_mQ

2

D

#

₀

1 1

_6m

1

₂

M M

ˆ

ˆ

since Q

.

_Q

1

_{(a) /(6m), and Q}

#

_{(a) /(2m), and hence (B) is established.}

_j

1 1 1

Proof of Lemma 5.

By Eq. (20) in the main text, or Eq. (A.13) above:

2

_ˆ

≠

_E[V(

u

_)]

₁

≠

u

≠u

]]]

5

E

_V

-

₍

u

₎

]

S DF

]]

2

]

G

_f(a

2

_{P ) dP}

2

_ˆ

_≠

_Q

_≠

_P

1 1

≠

_Q

₁

_h

1 1

2

_ˆ

1

≠

u

ˆ

]

]]]

1

E

_V

0

₍

u

₎

S

2

D

_f(a

2

_{P ) dP}

1

1 1

ˆ

≠

_P

≠

_Q

h

1 1

ˆ

ˆ

≠

_f(a

2

_{P )}

≠u

≠

₁

≠u

1

ˆ

]

]]

]

ˆ

]

]]]

E

V

0

₍

u

₎

S D

2

S D

_f(a

2

_{P ) dP}

1

E

_V

0

₍

u

₎

S D

2

_dP

1 1

≠

_P

≠

_Q

_ˆ

≠

_P

≠

_Q

1 1

h

1 1

(A.18)

By Eq. (A.2), the last term of the above expression is:

2

t

_Q

_h

t

_Q

1

≠u

1 1 2

ˆ

]

]

]]

]]

E

V

0

₍

u

₎

F

2

G

F

2

1

_(a

2

_P

2

_{mQ )}

G

_f(a

2

_{P ) dP}

1 1 1 1

2

ˆ

≠

_P

_ˆ

_ˆ

h

1

h

h

1

≠u

ˆ

]

]

E

V

0

₍

u

₎

F

2

G

_mf

9

_(a

2

_{P ) dP}

_(A.19)

1 1

ˆ

≠

_P

Integration of the last term of Eq. (A.19) by parts yields,

2

ˆ

ˆ

1

≠u

₁

≠u

ˆ

]

]

ˆ

]

]

2

E

_V

0

₍

u

₎

S D

2

_mf

9

_(a

2

_{P ) dP}

5

E

_V

-

₍

u

_{) m}

S D

_f(a

2

_{P ) dP}

1 1 1 1

ˆ

≠

_P

_ˆ

≠

_P

h

1

h

1

That together with the first term of Eq. (A.18) is

ˆ

ˆ

1

≠u

≠u

ˆ

]

]]

]

E

V

-

₍

u

₎

S DF

2

G

_f(a

2

_{P ) dP}

1

1 1

ˆ

≠

_Q

≠

_P

h

1 1

2

ˆ

1

≠u

ˆ

]

]

E

V

-

₍

u

₎

_m

S D

_f(a

2

_{P ) dP}

5

1 1

ˆ

≠

_P

h

1

ˆ

ˆ

ˆ

≠u

₁

≠u

≠u

ˆ

]

]

]]

]

E

V

-

₍

u

₎

F

2

G F

2

_m

G

_f(a

2

_{P ) dP}

5

1 1

≠

_P

_ˆ

≠

_Q

≠

_P

1

h

1 1

ˆ

t

_(a

2

_P

2

_{mQ )}

ˆ

≠u

₁

≠u

1 1

ˆ

]

]

]]]]]

]

ˆ

E

V

-

₍

u

₎

F

2

G F

1

₂

₍

u 2

_m)

G

_f(a

2

_{P ) dP}

1 1

≠

_P

_ˆ

_ˆ

≠

_P

1

h

h

1

(A.20)

and inserting this result in Eqs. (A.19) and (A.18) and rearranging,

2

_ˆ

_ˆ

_ˆ

≠

_E[V(

u

_)]

≠u

_{1 (}

u 2

_m)

ˆ

]]]

5

E

_V

-

₍

u

₎

F

2

]

G

] ]]]

_f(a

2

_{P ) dP}

1

2

_≠

_P

_ˆ

_Q

1 1

≠

_Q

₁ 1

h

1

ˆ

≠u

₁

≠u

ˆ

]

]

]

ˆ

2

E

V

-

₍

u

₎

F

2

G

S D

₍

u 2

_m)f(a

2

_{P ) dP}

1

1 1

≠

_P

_ˆ

≠

_P

1

h

1

2 2

t

_Q

2

t

_h

₁

1

ˆ

]]]

]

E

V

0

₍

u

₎

S

D

_f(a

2

_{P ) dP}

1

1 1

2

ˆ

ˆ

h

h

ˆ

_2Q

t

≠u

1

ˆ

]

]]

E

V

0

₍

u

₎

F

2

G

S

2

D

_f(a

2

_{P ) dP}

1

1 1

2

≠

_P

_ˆ

1

h

ˆ

ˆ

1

≠u

≠u

ˆ

]

]

]

E

V

0

₍

u

₎

S DS D

2

_f(a

2

_{P ) dP}

1

1 1

ˆ

≠

_P

≠

_P

h

1 1

2

ˆ

_h

t

_Q

1

≠u

1 ₂

ˆ

]

]

]]

E

V

0

₍

u

₎

S D

2

_s

_a

2

_P

2

_mQ

_d

_f(a

2

_{P ) dP}

_(A.21)

1 1 1 1

2

ˆ

≠

_P

_ˆ

h

1

h

2

ˆ

_h

t

_Q

1

≠u

1 2

ˆ

]

]

]]

E

V

0

₍

u

₎

S D

2

_(a

2

_P

2

_{mQ ) f(a}

2

_{P ) dP}

5

1 1 1 1

2

ˆ

≠

_P

_ˆ

h

1

h

ˆ

1

≠u

ˆ

]

]

ˆ

2

E

_V

0

₍

u

₎

S D

2

₍

u 2

_m)f

9

_(a

2

_{P ) dP}

5

1 1

ˆ

≠

_P

h

1

ˆ

ˆ

1

≠u

≠u

ˆ

]

]

]

ˆ

2

E

_V

-

₍

u

₎

S DS D

2

₍

u 2

_m)f

9

_(a

2

_{P ) dP}

1 1

ˆ

≠

_P

≠

_P

h

1 1

ˆ

ˆ

1

≠u

≠u

ˆ

]

]

]

2

E

_V

0

₍

u

₎

S DS D

2

_f(a

2

_{P ) dP}

_(A.22)

1 1

ˆ

≠

_P

≠

_P

h

1 1

inserting this result in Eq. (A.21) and rearranging:

2

_ˆ

_ˆ

_ˆ

≠

_E[V(

u

_)]

≠u

_{1 (}

u 2

_m)

ˆ

]]]

5

E

_V

-

₍

u

₎

S D

2

]

] ]]]

_f(a

2

_{P ) dP}

1

1 1

≠

_Q

≠

_P

_ˆ

_q

1 1

h

1

ˆ

ˆ

ˆ

≠u

₁

≠u

₁

≠u

₁

ˆ

]

]

]

ˆ

ˆ

] ] ]

E

V

-

₍

u

₎

S D S D

2

₍

u 2

_m)f(a

2

_{P ) dP}

2

E

_V

0

₍

u

₎

S D

_f(a

2

_{P ) dP}

1 1 1 1

≠

_P

_ˆ

≠

_P

_ˆ

≠

_P

_ˆ

1

h

1

h

1

h

ˆ

ˆ

t

_h

≠u

_{h (}

u 2

_m)

ˆ

]

ˆ

]

] ]]]

2

E

_V

0

₍

u

₎

_f(a

2

_{P ) dP}

5

E

_V

-

₍

u

₎

S D

2

_f(a

2

_{P ) dP}

1 1 1 1

3 2

ˆ

≠

_P

_ˆ

_Q

h

1

h

1

ˆ

1

≠u

ˆ

]]

]

2

E

_V

0

₍

u

₎

S D

2

_f(a

2

_{P ) dP}

_(A.23)

1 1

ˆ

dP

hQ

₁ 1

Now note that the sign of Eq. (A.23) is not clear since the sign of the first term of the

15

right hand side of it is the opposite to that the second term.

ˆ

ˆ

ˆ

ˆ ˆ

However, since V

-

₍

u

₎

5

_3V

0

₍

u

_{), or V}

0

₍

u

₎

5

_{(1 / 3)V}

-

₍

u

₎

u

_{, we can express Eq. (A.23)}

as:

2

_ˆ

_ˆ

_ˆ

≠

_E[V(

u

_)]

≠u

₁

_h

ˆ

FS D

ˆ

G

]]]

5

E

_V

-

₍

u

₎

S D

2

]

]]

_h

1

]

u 2

_mh

_f(a

2

_{P ) dP}

_(A.24)

2

≠

_P

_ˆ

2

₃

1 1

≠

_Q

₁

_{h Q}

1 1

15 _{ˆ ˆ}

In particular the sign of the first term of the r.h.s. is positive since V-(u)(2(≠u) /(≠P )) is a negative,1

ˆ

decreasing function of P , and (1 u 2m) is a function (linear) that goes form positive to negative as P goes1

from2`to1`, reaching the value of zero at the mean of P . But, the sign of the second term of Eq. (23) is1

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