FS IV 93 - 5
Information Sharing through Sales Report
Yong Yin
January 1993
ISSN Nr. 0722 - 6748
Forschungsschwerpunkt Marktprozeß und Unter
nehmensentwicklung (ÜMV) Research Unit
Market Processes and
Corporate Development (HM)
Paper FS IV 93 - 5, Wissenschaftszentrum Berlin, 1993.
Wissenschaftszentrum Berlin für Sozialforschung gGmbH, Reichpietschufer 50, W-1000 Berlin 30, Tel. (030) 2 54 91 - 0
Information Sharing through Sales Report
The paper studies information sharing through sales reports in duopoly with common demand uncertainty. Different from the previous information sharing model, firms report their past sales to a trade association rather than pooling signals about future demand. Our results differ from the previous ones regarding firms incentives for information sharing and the social welfare effect. Without long term commitment, sales reports are not self-enforcing. With long term commitment, sales reports make Cournot (Bertrand) firms better off when goods are substitutes (complements). Sales reports decrease (increase) consumer surplus and social welfare in a Cournot (Bertrand) industry.
ZUSAMMENFASSUNG
Marktinformationssysteme auf der Grundlage von Umsatzberichten
Diese Arbeit untersucht Marktinformationssysteme am Beispiel des Informationsaus
tauschs zwischen duopolistischen Unternehmen bei gemeinsamer Nachfrageunsicher
heit. Im Unterschied zu herkömmlichen Modellen des Informationsaustauschs ziehen die Unternehmen es hier vor, ihre Umsatzberichte an einen Wirtschaftsverband zu übermitteln, anstatt Informationen über die mögliche zukünftige Nachfrageentwick
lung auszutauschen. Die vorliegenden Ergebnisse unterscheiden sich von früheren sowohl in bezug auf die Anreize für Unternehmen Informationen auszutauschen, als auch in bezug auf den Effekt auf die gesellschaftliche Wohlfahrt. Ohne langfristige Verpflichtung sind Umsatzberichte ohne Belang. Mit langfristiger Verpflichtung ver
bessern Umsatzberichte die Situation von Unternehmen im Cournot (Bertrand) Fall, wenn es sich bei den produzierten Gütern um substitutive Güter (komplementäre Gü
ter) handelt. Umsatzberichte senken (steigern) die Konsumentenrente und die ge
sellschaftliche Wohlfahrt in einer Cournot (Bertrand) Industrie.
1. Introduction
We study the effect of sales reports in duopoly with common demand uncertainty. The results we get differ from the previous information sharing model with signal pooling. If firms can not resist short run temptation, sales reports are not self-enforcing. With long term commitment, sales reports are compatible with non-cooperative behaviour in a Cournot (Bertrand) industry with substitute (complement) products. Sales reports decrease (increase) consumer surplus and social welfare in a Cournot (Bertrand) industry.
The information sharing problem has become an interesting topic in economic literature. Studies have been focusing on firms' incentives to share information and its social welfare effect. Previous models (Novshek and Sonnenschein [1982], Clarke [1983], Vives [1984], Gal-Or [1985], [1986] among others) have shown that firms' incentives depend on various industry characteristics, such as the type of competition (Coumot/Bertrand), the feature of products (complement/substitute) and so on. In particular, in a Cournot (Bertrand) model with common demand uncertainty, information sharing is a subgame perfect equilibrium if and only if products are complements (substitutes). In a Cournot oligopoly with homogeneous products, Kirby (1988) found that the result can be reversed if costs are convex and information sharing is exclusive. Hviid [1989], Sakai and Yoshizumi (1991), working on a similar model, also gave different results when firms are risk-averse.
All these models assumed that firms share signals about future demand. However in
many industries firms often report past sales or price data to trade associations (Vives
[1990]). Different from signal pooling, sales/price reports enable firms to look back what
others have done. In an antitrust case against American Column and Lumber Co. in 1919, the
trade association, in defending its conduct in the court, claimed (Baldwin [1987]): "The Open
Competition Plan is a central clearing house for information on prices, trade statistics and
practices. By keeping all members fully and quickly informed of what others have done, the
work of the Plan results in a certain uniformity of trade practice. There is no agreement to
follow the practice o f others, although members do follow their most intelligent competitors, if they know w hat these competitors have been actually doing."
Economists often suspect that firms use sales/price reports to facilitate collusion. Salop [1986] wrote: "Information exchange facilitates both explicit and tacit co-ordination by eliminating uncertainty about rivals' actions. Classic examples o f information exchanges are inter-seller verification o f price quotations and advance notice o f price changes." Generally speaking, courts view past sales/price reports less collusion conducive than advance notice o f price changes. In the U.S. and most western countries, "the courts have held that trade associations must report only past and current prices and production rates. In particular, advanced notice o f a price change cannot be disseminated to competitors through the trade association" (Baldwin [1987]). However, an antitrust authority may cross the line and further challenge past data sharing. In a recent case, the European Commission banned a group o f agricultural tractor suppliers in the United Kingdom from sharing past sales. The data were provided by a private company and sold to the members o f an information exchange system.
In EC's point o f view, the information exchange reduces firms' incentives for price cutting and hence is collusion conducive. M ost members o f the system, including international conglomerates as well as domestic manufacturers, appeal to the court defending their motivations as non-collusive. The decision has yet be made by the court.
In dealing with this type o f problems, one finds that price/sales reports are not equivalent to signal pooling. Although both past data and signals provide information about the future, the form er can be affected by firms' previous decisions while the latter can not.
Under demand uncertainty without sales reports, firms' past prices or outputs are not observed by rivals. This gives an incentive to manipulate rivals' perception about the market. A firm can choose its output/price different from the current profit maximisation level. Information manipulation, even though it may not successfully fool the rival, will change both firms' payoffs. Sales reports eliminate such an information manipulation incentive and keep firms maximising current payoffs. Firms' incentives for sales reporting depend on whether it
increases their expected profits. These incentives are not the same as the incentives for
sign alpooling. Recently Mirman, Samuelson and Schlee [1992] studied the information manipulation problem in a Cournot duopoly model. They considered
in form ationmanipulation as to change "the extend to which belief revision occurs rather than the direction in which beliefs are revised." Their question is whether firms want the market price to be more informative. Our question though, is whether firms prefer information manipulation (without sales reports) or information exchange (through sales reports). We also evaluate the social welfare effect of sales reports.
2. The Model
We consider a two period symmetric Cournot or Bertrand duopoly model with linear demand. For simplicity we let costs be zero. Consumers' utility in period t is
1
(1) x ot + a t(x l t + x 2t) - 2 b ( x l t 2 + x 2t2) + 0 b x l t x 2t> t = 1,2.
where b and 0 are constant and known. 0 E [-1,1], if 0 > 0, products are complements, if 0 <
0 products are substitutes. x0 is a composite good with unit marginal utility. xit is firm i's product at time t. Consumers' utility maximisation results in linear (inverse) demand functions pjt = at - bx;t + 0bxjt in a Cournot case and Xjt = a t - ßp; + yßpj in a Bertrand case, where a t = b(l-0)’ H v ß = b(l-02) 1 and y = -0. Firm i's profit function in period t is jtit = (at - bxj,
+ 0bxit)xit in a Cournot model and Jtit =
(a t -ßpit +
yßPjt)Pit *n aBertrand model. In a Cournot case, social welfare and consumer surplus are functions of firms' outputs. We can write them as
1
(2) SW = at(xlt + x2t) - 2b(xlt2 + x2 2) + 0bxltx2t (3) CS = |b (x lt2 + x2t2) - 0 b x ltx2t
In a Bertrand case, social welfare and consumer surplus are functions of firms' prices.
(2') SW ' = ß O ^ ) + VßPltP2t - jß (P lt2 + P2t2)
Q 2 *£
(3') CS’ = ß ( H ) ' yßPltP2t + 2^(Plt2 + P2t2) - «t(Pit + P2t)
There is a strong similarity between a Cournot game and a Bertrand game. Ignoring parts which are independent o f firms' decisions, social welfare in one model has the same functional form as consumer surplus in the other model w ith the opposite sign. To save spaces in the following we will deal with the Cournot model and translate our conclusion for the Bertrand case with appropriate modification.
W e assume that a demand sequence is governed by a M arkov process: at = at.x + et where Etat = 0. There is no information about et at time t. The most accurate estimate o f at at period t is the value o f at_x. W ithout sales reports, each firm only knows its own sales and price. A firm can report its sales to its rival through a trade association. W e assume that data are verifiable, so cheating is ruled out. W hen a Cournot firm reports its sales in period one x;i, its rival can figure out the demand a! before it takes action in period two. Alternatively, if a firm reports its price p;i, its rival can also calculate ax from the two demand functions. Thus sales reports and price reports are equivalent. This equivalency holds because there is only one random variable. If two firms face different demand shocks, price reports and sales reports should be treated differently. Bearing this in mind, we w ill only refer sales reports in this paper.
In period one, firms know a0 and maximise their expected profits from two periods. In period two they maximise expected profits in that period. With sales reports, they can find out a! before they take actions in period two. There is no chance to manipulate a rival's belief by choosing different quantities in the first period. Hence in the first period a firm only needs to maximise the current profit. W ithout sales reports, firms can not directly calculate ax. They have to rely on their own price and sales to infer ax. In this case a firm has an incentive to deviate from the current profit maximisation to change its rival's perception about demand. In so doing, it may induce its rival's action in its benefit. For instance, w ith substitute products,
one firm is always better o ff when its rival produces less. If a firm produces more in period one, its rival sees a lower price. Anticipating a low demand from the low price, the rival would produce less in the second period. Therefore firms tend to produce more in the first period if sales are not reported.
W e use a subgame perfect rational equilibrium where firms' strategies in both periods are the best responses to each other and in period one their expectations o f the second period actions are rational. It implies that firms can not fool each other w ith information manipulation. In period two firms are able to rationally infer the true value o f ax even though there is no sales report. However, firms have to manipulate information in order to distinguish from the w orst case. If information manipulation leads to a more competitive situation, sales reports, which eliminates such manipulation, will have the opposite effect, i.e.
collusion conducive. In this case we should discourage sales reports. Otherwise, no intervention is needed.
3. Equilibria under Different Sales Report Schemes
We will discuss three different cases: (1) "dual reports", both firms report sales; (2)
"unilateral report", only one firm reports and (3) "no report". W e find the subgame perfect equilibrium strategies for each o f the three cases. Under different sales report schemes, firms' profits, consumer surplus and social welfare are different in the first period. But the second period equilibrium outcomes will not change. Therefore we evaluate profits, consumer surplus and social welfare in period one under different sales report schemes. Comparing them, w e can evaluate firms' incentives for sales reports and the social welfare effect.
3.1 D ual Reports
W hen both firms' sales are reported, firms are able to figure out the true value o f ax in period two. There is no incentive to deviate from the current profit maximization in period one. Firms choose first period outputs, x xl and x2X to m axim ise Jtlx and given a0. In the
second period firms' decisions depend on their own prices and both firms' quantities in the first period. W e find firms' optimal strategies as xix = and xi2 =
Under these strategies, firms' expected profits in period one are
ma nr - —a»2
(4) ^ii (2 -0)2b
Expected social welfare and consumer surplus are (5)
(6)
SW =
CS =
(3-9)a02 (2-0)2b ( W
(2-0)2b ‘ 3.2 Unilateral Report
Under this scheme firm 2's sales are reported, but not firm l's. Firm 2 can not directly find av Its quantity decision in period two, x22, can only depend on its own price and sales in period one. Its objective in period one is simply to maximise its current profit. There is no point to deviate from the myopic position because firm 1 can find out ax. However firm 1 has an incentive to deviate. Its decision in period two, x 12 depends on p n , xn and x21. W hile firm 1 has the chance to manipulate firm 2's information, firm 2 is not fooled and is able to choose x22 rationally and optimally. This situation resembles Spence's signalling gam e (1974) with separating equilibrium. However, in Spence's model the signal sender bears the cost o f signalling, while in our case, both parties are affected directly. W ith substitute products, when firm 1 sends a low demand signal by overproducing, it imposes a damage on firm 2.
W ith linear demand, firm 2's price p21 is linear in ax and x21. Firm 2's optimal output in the second period, x22, is linear in p21 and x21. W e denote it by + ß2x2l + Y2P21 w h e r e p,2, ß2 and Y2 are undetermined constants. Let A. E [0,1] be firms' discount factor for the second period profit. W ith the subgame perfect equilibrium conditions, we can find firms' equilibrium strategies (see Appendix A). We get
Proposition 1: The equilibrium strategies with firm 2's unilateral report are
X11 ;U +
2X02
(2 -0 )b ' (2-0)2(2+0) X21 =
äü ;{i + X03 (2-0)bT (2-0)2(2+0) _ P11 + b x n -b 0 x ?1
Xl2 “ (2-0)b x 22 =
bx91+p91 0an
{!+■ 2X02
(2-0)b (2-0)2b v T (2-0)2(2+0)
} }
Compared with the dual report case, firm 1 always overproduces in period one under unilateral report, while firm 2 overproduces if and only if goods are complements.
Translating this into a Bertrand model, one gets that firm 1 always charges a higher price in period one and firm 2 charges a higher price if and only if goods are substitutes.
Overproduction would diminish if X approaches zero. Based on the equilibrium strategies above, it is easy to show that firms' first period profits as:
(7) „ ... an2 (4-02)(2-e-X)+X02
11 (2-0)2b { (2-0)4(2+0)2 }
an2 XQ3 2
7121 " (2-0)2b ^ 1+(2-0)2(2+0)^
(8)
One can see the differences between the profits in the unilateral report case and those in the dual report case. We find that (7) is always greater than (4), while (8) is greater than (4) if and only if 0 is positive. In a Bertrand model, the same must hold true with 0 replaced by y.
Therefore when a Cournot (Bertrand) firm withholds sales data, relative to the duel report case, its output (price) and profit are higher, while its rival's output (price) and profit are higher if and only if goods are complements (substitutes). W e can numerically show the magnitude o f the effect o f sales reports for some particular values o f 0 and X. Let 0 = -1 (homogeneous products) and X = 0.9. W hen firm 1 withholds its sales, its profit increases by 8%. But firm 2 loses 19% of its profit. In a Bertrand case with y = 1 (strong substitutes) and X
= 0.9, firm l's profit rises by 12%. Firm 2 gains by 69%, making a higher profit than firm 1 does.
In order to consider firms' incentives for sales reporting, we allow firms simultaneously decide w hether to report sales at the beginning. From the above, we see that given one firm reports sales, the other firm always prefers not to report. Hence dual reports are never sustainable as a subgame perfect equilibrium. The previous result says that sharing demand
signals is sustainable in a Cournot (Bertrand) model with complement (substitute) products.
The conclusions with sales reporting and w ith signal sharing are different. They are sensitive to w hat kind o f information is shared.
3.3 No Report
W e study the case where both firms withhold sales data. Firms' outputs in period two depend on their own prices and outputs in period one. W e denote firms' optimal decision rule in the second period by xi2 = p + ßxix + ypp Firms can not directly calculate a1, however by our assumption, in the equilibrium they act rationally. Even though information is manipulated, the true state o f demand is still revealed in the equilibrium. W e find firms' equilibrium strategies in period one and period two as follows (see Appendix B).
Proposition
2:
W ithout sales reports firms' equilibrium strategies areÄ02 0an 0 X02 (2+02)xi1 Eil
Xil (2 - 0 )b ^ + (2-0)2^ and Xi2 ' (2-0)2b('1 + 2 ^ 1 + (2-0)2^ + 2(2-0) + (2-0)b' Firms overproduce in period one compared with the dual report case. For a Bertrand industry, firms charge higher prices without sales reports. Given the equilibrium strategies, the profit in period one can be shown as
™ „ an2 .. + ln3(2-8)2-M(l-8)
(9) » il (2-0)2tJ M (2-8)4 }
A n interesting question is whether a firm has incentive to unilaterally report its sales given that its rival withholds sales data. To answer the question, we compare (9) w ith (8). We find that the form er is always greater than the latter. So a firm has no incentive to report its sales without reciprocity. It was shown earlier, that given its rival reports sales, a firm has no incentive to join in. Thus, regardless o f what its rival does, a firm always prefers not to report its sales. If firms can not co-ordinate sales reports for the long term relationship and assume that rival's decisions are fixed, no sales report is a dominant strategy. W e thus get
Proposition 3: If firms make sales report decisions independently, no sales report is the only subgame perfect equilibrium.
The proposition says that sales reports can not be firms' independent choices without any co-ordination based on long term relationship. The relationship reflects f ir m s ' interdependency, it may or may not be associated with collusion. In the most cases a sales reporting system can only be viable when both firms participate. They can not independently make decisions on sales reports in long run. I f firms realise the benefit o f sales reports, they may set up a sales reporting system jointly. This situation is not the same as in a quantity or price game. Hence, the comparison we made above could be misleading. To determine whether a sales reporting system is viable, it might be more appropriate to compare firms' profits with dual reports and with no report, i.e. (4) vs. (9). W e find that the former is greater than the latter if and only if 0 < 0. Thus sales reports benefit Cournot firms if and only if goods are substitutes. Bertrand firms' profits increase w ith sales reports if and only if goods are complements. Hence we have
Proposition 4: Sales reports increase Cournot (Bertrand) firms' profits if and only if
goods are substitutes (complements).
Even if firms are capable o f making long term commitment, they would not voluntarily set up a sales reporting system in a Cournot (Bertrand) industry w ith complement (substitute) products. Such a system would be set up in a Cournot (Bertrand) model with substitute (complement) products. This is very different from what the previous signal sharing model, which says that Cournot (Bertrand) firms are better o ff with signal sharing if goods are not strong substitutes (complements). In our Cournot case, when goods are homogeneous and X is 0.9, sales reports increase firms' profits by 17%. In a Bertrand model w ith y = 1 (strong substitutes) and X = 0.9, sales reports decrease firms' profits by 46%.
W e then evaluate the effect o f sales reports on consumer surplus and social welfare.
Substituting our equilibrium strategies without sales reports into (2) and (3), we find the expected social welfare and consumer surplus in period one as
(10) an2 X02
SW “ (2-0)2b ^ + (2-0)2} { 3 - 0 X02(l-0 ) (2-0)2
(11) c s _ a o M
(2-0)2b 1 (2-0}
äl2They are greater than (5) and (6) respectively. Hence sales reports always hurt consumers and the society in a Cournot industry. With 0 = -1 and X = 0.9, sales reports reduce consumer surplus by 17%. The social welfare loss is about 5%. For a Bertrand case, the conclusion is obtained by similarity argument as before. W e get
Proposition 5: Sales reports reduce (raise) consumer surplus and social welfare in a
Cournot (Bertrand) industry.
In a Cournot model sales reports always lead to production restraint. W hen products are complements, production restraint means higher prices, but also weakens the positive externality between firms. Since the latter is dominant, firms get lower profits w ith sales reports. W ith substitute products, sales reports reduce competition and benefit firms at expenses o f consumers. In such a case sales reports result in a collusive outcome without collusion and should not be left free o f intervention. In a Bertrand case, sales reports always reduce firms' incentives to raise prices and bring about lower prices. However, with complements, the effect o f low prices is dominated by stronger positive externality. Sales reports benefit firms as well as consumers and no government intervention is needed. With substitute products, sales reports induce more price cuts and enhance competition. Firms do not have the incentive to do so, even though it is socially desirable. Government may step in and force them to do so.
4. Concluding Remarks
Having examined the information sharing problem with sales reports, we find some interesting results regarding firms' incentives and social welfare. If firms are short-sighted, sales reports can not arise as non-collusive firms' voluntary independent decisions. When firms can tie themselves for the long term interests, sales reports are rational for Cournot (Bertrand) firms w ith substitute (complement) products. In a Cournot (Bertrand) industry,
sales reports decrease (increase) consumer surplus and social welfare. These results are very different from w hat we saw in the signal sharing model. We need to pay serious attention to what kind o f information is shared as we justify firms' incentives for information sharing and study its social welfare consequences. In the real world, information sharing may involve factors o f both signal pooling and sales reporting. The question o f which model is more appropriate depends on the detailed specification o f an particular industry.
There are certain shortcomings in this model. W e assumed that firms perfectly know a0.
in period one. This is crucial for them to rationally infer the value o f a-i in period two without sales reports. It would be more reasonable to assume that firms know a0 imperfectly. Then firms' perfect inference about a T becomes impossible without sales reports. Therefore sales reports indeed provide firms more accurate information about demand. This will reduce firms' costs in terms o f inventory and forgone consumers. In this case sales reports would have extra values which are not considered in our model. This point should be taken into account. For instance, in a Bertrand industry with substitute products, firms' gain from m ore accurate information may overweigh the loss we calculate in our model. Then non-cooperative firms may have incentives for sales reports.
Appendix A, P ro o f o f Proposition 1:
As Ea1 = a0, the first order condition o f x21 is (A l) a0 - 2bx21 + 00X2! = 0
The first order condition o f x u is
(A2) “« - 2bxu + 0bx2l + WE(aXi2äXu + ax22äXn) = 0
0X22 _
r 3%f9* .
^X22
a0 - 2bxn + b0x21 + Xy202b2E x12 = 0
j u v v v j u a i IU
any al5 we must have y2 = g ^ - Rational expectation implies that E x12 =
The first order condition o f x 12 implies E (~ ) = 0. Since x22 = a 2 + ß2X2i + Y2P2I’ =
dx12 dxn
y20b. W e also know = 0bx12. Therefore w e can write (A2) as (A3)
If x22 is indeed optimal, it should be equal to a 2 + ß2x 2 i + Y2P21 = (2 0 )b ^o r Hence
(z-ojD * * “ (z-ö;d
(A3) becomes
(A4) a0 - 2bxn + b0x21 + = 0
Solving (A l) and (A4), we get
(A5) 2X02
11 (2 - 0 )b ^ + (2-0)2(2+0) Xu =■
(A6) ____ X03
X21 “ (2 -0 )b ^ + (2-0)2(2+0)^JkL
Now we consider firm 2's second period output, x22. It should not be affected by its first period output, x21. Thus 7“ 1 = 0, which implies ß2 = . Substituting x u , x21 i
dx21 (^-v)
a! 0an 2X02
function o f x22 and let it be equal to we get 02 = ’(2-0)2b ^ + (2-0)2(2+0)^'
into the }
1
Appendix B, P ro o f o f Proposition 2:
The first order condition o f firm i's output in period one, x;i is (B l) a o . 2bX il+ebX jl + XE(» + « ) } = o
n , dxi2 • „z^JtpdXp „ , dJTjo dXr9
Since E fc -^ ) = 0 and ■“ is constant, E(' - lZ_ u = 0. Substituting — 12 = 0bxi9 and — =
^xi2 dXii dxi2dXil 0 dxi2 12 dXil
Y ^ = Yeb , we can write (B l) as 'ö x L
(B2) a0 - 2bxil + Obx^ + Xy02b2Exi2 = 0
We can solve two equations o f (B2) for i = 1 and 2. We get
(B3) xil = (4-02)b {2 tao + ^Y02b2Exi2] + 0[ao + X y O ^ E x ^ ]}
_ _ ä i Ai2 1
By the optimality condition, xi2 = ■ If the rule xi2 = a + ßx;i + yp;i is indeed optimal, y
1 a
must be equal to W ith rational expectation Exi2 = Exj2 = Substituting these
into (B3), we get
(B4)
02
X;1 (2 - 0 )b ^ +X ( 2 - 0 ) ^
Now we consider xi2. W e know that x;i affects xi2 only through Xj2. But x.-2 is affected by x ^
. x. _ , dxi9 n 1 dxi9dxi9 0ydpji 02
only through Pjl. Thus we have = ß - — = = 2 ^ =
1 .. 92. ... ... aj Hence we must have ß = 7 7 7 7 (1 + 77). Substituting it into xi2 and using optimality condition, xi2 =
~ 1
(Z -u ) z (2 -0 )b
0an 0 2X02
find a - -(2.0)2b( 1 + + (2-0)2^' H
we
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