Discussion Paper No. 752

RELATIVE PERFORMANCE AND
R&D COMPETITION
Toshihiro Matsumura
Noriaki Matsushima
Susumu Cato

August 2009

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Relative Performance and R&D Competition

Toshihiro Matsumura
Institute of Social Science, University of Tokyo
Noriaki Matsushima∗
Institute of Social and Economic Research, Osaka University
andSusumu Cato
Institute of Social Science, University of Tokyo.
August 24, 2009

Abstract
This paper formulates a duopoly model in which ﬁrms care about relative proﬁts as wellas their own proﬁts. Our purpose is to investigate the relationship between the weight ofrelative performance and R&D expenditure. We ﬁnd a non-monotone relationship between
the weight of relative performance in their objectives and their R&D levels. Both highly
reciprocal (altruism) and negative reciprocal attitudes yield high levels of R&D, while theintermediate situations yield low levels of R&D.

JEL classiﬁcation numbers: L21, O31, L13
Key words: relative proﬁt, innovation, spite, altruism∗
Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University,
Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6877-5111 (ex. 9170), Fax: +81-6-6878-2766,
E-mail: nmatsush@iser.osaka-u.ac.jp1

1

Introduction

One tends to care about the performance of other people as well as one’s own performance.
This concern may either stem from the available incentive schemes or from just one’s intrinsic
interest. For instance, evaluations of managers’ performances are often based on their relative
performance as well as their absolute performance (Murphy (1998)). Outperforming managers
often obtain good positions in management job markets. In this case, these managers act in a
way that allows them to meet the relevant incentive schemes. Moreover, a considerable amountof laboratory (experimental) research has pointed out spiteful behavior as well as reciprocal
or altruistic behavior, which is closely related to the objective functions of agents based onrelative performance (Brandts et al. (2004), Cason et al. (2002), and Coats and Neilson (2005)).
These reciprocal and spiteful preferences often stem from genuine emotions or incentive schemes.
Therefore, incorporating these preferences into a model is an important research topic.
The purpose of this paper is to investigate the relationship between the payoﬀ functions ofﬁrms incorporating both positive and negative reciprocal preferences and the R&D expendituresof the ﬁrms. The outline of the model treated here is as follows. Firm i’s payoﬀ is its relative
proﬁt πi − απj , where πi is its own proﬁt, πj is the rival’s proﬁt, and α ∈ (−1, 1). The parameterα represents the degree of reciprocal preference. If α is positive, the ﬁrms envy the rivals’
success. If α is negative, the ﬁrms have reciprocal (altruism) payoﬀ functions.1
We ﬁnd a non-monotone (U-shaped) relationship between the degree of reciprocal preference
α and the innovation activities. Given a nonpositive α, an increase in α reduces R&D. When αreaches a critical value (which is strictly larger than 0), the relationship is inverted. From thecritical value of α, an increase in α increases the levels of R&D. This result indicates that R&D
activities are quite active in both highly cooperative (α is close to −1) and highly noncooperative
1

The parameter α is closely related to the “coeﬃcient of eﬀective sympathy” used by Edgeworth (1881)and “coeﬃcient of cooperation” used by Cyert and DeGroot (1973). For discussions on non-proﬁt maximizing
preferences in this context, see Bolton and Ockenfels (2000) and Fehr and Schmidt (1999).

2

(α is close to 1) societies, while they are less active in the intermediate cases.2
Through the direct interpretation of α, our result sheds some light on how reciprocal andnegative reciprocal attitudes aﬀect R&D. Given the pure selﬁsh preferences of ﬁrms, an introduction of minor spiteful preferences reduces their innovative activities, but an introduction ofsigniﬁcant spiteful preferences stimulates their innovation.3
We extend our basic analysis in two directions. First, we consider a joint R&D implementation where ﬁrms cooperatively choose their R&D levels and then compete in the productmarkets. In general, collusion in the product market is per se illegal, while it is possible for
R&D cooperation to be allowed. Thus, this situation is worth discussing. It is shown that inthis situation, an increase in α reduces R&D. Second, an oligopoly model is considered. It isshown that an increase in α is less likely to stimulate R&D when the number of ﬁrms is larger.
We add some comments on rationales for discussing objective functions on the basis of relative performance. First, relative performance, especially in a positive α case, is quite importantfrom the viewpoint of evolutionary stability.4 Second, owners of ﬁrms often have incentives for
adopting a positive α in the context of strategic commitment games (Kockesen et al., 2000).
2

Although the literature on strategic R&D competition is fairly abundant, most papers assume Cournot

competition, where ﬁrms maximize their own proﬁts. See, among others, Brander and Spencer (1983), Spence(1984), d’Aspremont and Jacquemin (1988), Suzumura (1992), Kamien et al. (1992), Matsumura (1995), andLahiri and Ono (1999).
3

The payoﬀ functions that are based on relative wage or relative wealth status have also been intensively

discussed in the macroeconomics context. Keynes (1936) discussed the rigidity of nominal wage based on relative
wage. See also Akerlof and Yellen (1988), Corneo and Jeanne (1997, 1999), and Futagami and Shibata (1998).
The relative performance approach is important in political science. Obviously, a party cares about the numberof votes obtained not in absolute terms but in relative terms. In addition, in the context of international policies,
the possibility that governments care about their relative performance as well as absolute performance is pointed
out. See, among others, Grieco et al. (1993) and Mastanduno (1991).
4

See Alchian (1950) and Vega-Redondo (1997). Vega-Redondo (1997) also shows that Cournot competition

with relative performance objectives yields the Bertrand outcome even in duopoly, and this outcome is evolutionary
stable.

3

Third, as mentioned in Symeonidis (2008), we can interpret α as a parameter indicating severityof competition.5 α = 0 indicates the standard Cournot case; α = 1, the perfectly competitive
case (related to the Bertrand case); and α = −1, the monopoly case. Thus, a larger α indicatesa more competitive market.6 The relative performance approach enables us to treat competitiveness as a continuous variable, and this model contains three standard models—Cournot,
Bertrand, and monopoly—as special cases. We believe that our formulation has suﬃcient importance for an investigation into innovations and that our relative proﬁt approach is applicableto many other problems.
The rest of this paper is organized as follows. Section 2 formulates our basic model. In
Section 3, we demonstrate the U-shaped relationship between α and R&D expenditures. Section
4 provides the analysis of joint R&D implementation. In Section 5, the basic model is extended
to the case of oligopoly. We discuss welfare implications in Section 6. Section 7 concludes thispaper. All proofs are relegated to the Appendix.

2

The Basic Model

We formulate a two-stage symmetric duopoly model. In the ﬁrst stage, ﬁrm i (i = 1, 2) chooses
its R&D level Ii . At the beginning of the second stage, each ﬁrm observes the rival’s R&D.
In the second stage, ﬁrms produce perfectly substitutable commodities for which the market
demand function is given by p = a − Y (price as a function of quantity), where Y is the total
output of the ﬁrms. Let yi denote the output of ﬁrm i. Firm i’s marginal production cost ci
depends on Ii . Each ﬁrm i chooses yi independently.
The payoﬀ of ﬁrm i (i = 1, 2) is given by Ui = πi − απj (i ̸= j), where πi is the proﬁt of ﬁrm
5

See also Shubik (1980), Brod and Shivakumar (1999), and Symeonidis (2000).

6

Under the standard conditions in Cournot, the ratio of the proﬁt margin (the price minus the marginal cost)

and the price, called the Lerner index, is decreasing in α. This index is intensively used in the empirical literature
as a measure of competitiveness in product markets.

4

i and α ∈ (−1, 1). The parameter α indicates the importance of relative performance for ﬁrm
i’s management. The ﬁrm i’s proﬁt πi is given by πi = (a − Y )yi − ci (Ii )yi − Ii . It is assumed
that c′i ≤ 0 and c′′i is positive and suﬃciently large so as to satisfy the second-order condition at
the ﬁrst stage. We also assume that limI→0 c′ (I) = −∞ and limI→∞ c′ (I) = 0 so as to ensure
the interior solution at the ﬁrst stage.

3

Equilibrium Analysis and the U-shaped RelationshipIn this section, we study a situation where two ﬁrms maximizing relative proﬁts compete in themarket. The game is solved by backward induction. In the second stage competition, given theinvestments Ii of two ﬁrms, each ﬁrm independently chooses its output to maximize the relative
proﬁt Ui . The ﬁrst-order condition is as follows:a − 2yi − (1 − α)yj = ci(i = 1, 2, i ̸= j).

(1)

Obviously, the second-order condition is satisﬁed. Arranging this equation, we obtain the following reaction function:yi = Ri (yj : α) =a − ci − (1 − α)yj
.
2

(2)

By solving the ﬁrst-order condition, the second stage equilibrium outputs are obtained:y1E =(1 + α)a − 2c1 + (1 − α)c2
(1 + α)a − 2c2 + (1 − α)c1
, y2E =.
(1 + α)(3 − α)(1 + α)(3 − α)(3)The resulting proﬁt of ﬁrm i is given by
πiE (ci , cj ) =[(1 − α)a − (2 − α)ci + cj ][(1 + α)a − 2ci + (1 − α)cj ]− Ii .
(3 − α)2 (1 + α)

(4)Next, we consider the ﬁrst stage R&D competition. In this stage, each ﬁrm i independently
chooses Ii so as to maximize Ui = πiE − απjE . We restrict our attention to the symmetric5equilibrium because the suﬃciently large c′′i guarantees that the unique equilibrium is symmetric.
The ﬁrst-order condition is−[(1 + α)a − 2ci + (1 − α)cj ][4 − 3α + α2 ] ′ci = 1.
(3 − α)2 (1 + α)

(5)The second-order condition is satisﬁed. On substituting c1 = c2 = c into (5), we have that
−G(α)c′ = 1 must be satisﬁed, whereG(α) :=(a − c)(4 − 3α + α2 ).
(3 − α)2

Let I E denote the equilibrium R&D investment level. A larger G and a larger |c′ | imply a higher
marginal beneﬁt of R&D. Since the payoﬀ function is assumed to be concave with respect to Ii(c′′ is large enough), a larger G (as well as larger |c′ |) yields a higher level of the equilibrium
R&D investment.
We discuss how α aﬀects the equilibrium R&D level. We ﬁnd a non-monotone relationshipbetween the equilibrium level of R&D (I E ) and the weight of relative performance (α).
Proposition 1 Suppose that two ﬁrms compete in a product market, and they make their R&D
investments independently. Then, the equilibrium R&D investment level I E is decreasing in α
for α < 1/3 and is increasing in α for α > 1/3.
We explain the intuition behind Proposition 1.

Since yiE is increasing in α, the cost-

minimizing level of R&D, which is derived by minimizing ci yiE + Ii , is increasing in α. Thus,
for the purpose of cost minimization (minimization of production cost plus R&D cost), ﬁrm i
has a stronger incentive for R&D when α is larger. On the other hand, ﬁrm i has a weaker
incentive for innovation for strategic purposes.7 From (2), we have that |Ri′ | is decreasing in α.
This implies that the strategic value of R&D is decreasing in α. The former dominates when
7

From (3) we see that an increase in Ii (a decrease in ci ) decreases yjE , and it results in an increase in πi(i, j ∈ {1, 2}, i ̸= j). This is the strategic value of R&D. For this strategic eﬀect of R&D, see Brander and Spencer(1983).

6α is large, while the latter dominates when α is small. This yields the U-shaped relationshipbetween α and the equilibrium R&D level.
The implications of Proposition 1 are as follows. Both aggressive competition (the case where
α is close to 1) and a collusive situation (the case where α is close to −1) yield high levels of
R&D.8 Our result is related to two inﬂuential views on the relationship between competitivenessand R&D.9 One view is the monopoly view. The monopoly yields intensive R&D investments.
The other is the competition view. Severe competition accelerates innovation. Researchers havepresented many theoretical foundations and empirical (or anecdotal) evidences supporting bothviews. Using a single model, we explain that both views can be accurate.
In our setting, R&D is minimized when α = 1/3. Starting Cournot competition (α = 0),a slight increase in α decreases R&D investments, while a large increase in α increases them.
Thus, an envy society (positive α) in which people care about their relative performances aswell as their absolute performances can yield either more or less aggressive R&D activities.

4

Joint R&D Implementation

The question is whether or not our result depends on the assumption about the formation of
R&D activities. Then, we consider the case in which two ﬁrms10 cooperatively determine theirinvestment level I to maximize their joint proﬁts, while they noncooperatively compete in the8

For the relationship between α and market competition, see the rationale in the second last paragraph of theintroduction.
9

Economists have long been interested in the relationship between product market competition and innovation.

See Aghion et al. (2005). They show an inverse U-shaped relationship between toughness of competition and theequilibrium level of R&D. Traditionally, many researchers believe that monopoly yields intensive R&D (monopolyview), while many others believe that competition yields intensive R&D (competition view). Both have presented
many theoretical foundations and empirical (or anecdotal) evidences supporting their views. See, for example,
Schumpeter (1950) and Arrow (1962). See also Mateus and Moreira (2007), Cabral (2000), and the works citedin these books.
10

We can show that Proposition 2 holds true in n-ﬁrm oligopoly too.

7

product market.11 Some points of the basic model are modiﬁed. Each ﬁrm has a common
marginal cost d(I) that depends on the joint R&D investment I. We assume that each ﬁrm pays
half of the joint R&D cost I. Thus, ﬁrm i’s proﬁt is given byIΠi = (a − Y )yi − d(I)yi − .
2It is assumed that d′ (I) < 0 and d′′ (I) is positive and suﬃciently large. We also assume that
limI→0 d′ (I) = −∞ and limI→∞ d′ (I) = 0 so as to ensure the interior solution at the ﬁrst stage.
We consider a two-stage game. In the ﬁrst stage, the ﬁrms choose their R&D level cooperatively. In the second stage, they noncooperatively produce perfectly substitutable commodities.
We now solve the game. The second stage competition has already been discussed in theprevious section: all that needs to be done now is replace c(I) with d(I). In the ﬁrst stage, theﬁrms choose the investment level I. The ﬁrst-order condition is as follows:−2(a − d)(1 − α) ′d (I) = 1.
(3 − α)2(6)The left-hand side in (6) is the marginal beneﬁt of their joint R&D investment and the righthand side in (6) is the marginal cost of the investment. Let I C denote the equilibrium R&D
investment level. Deﬁne H(α) := 2(a − d)(1 − α)/(3 − α)2 . Because of the concavity of the payoﬀ
function, which is guaranteed by the assumption of suﬃciently large d′′ , a larger H implies ahigher level of investment. We investigate how α aﬀects the equilibrium R&D level. In contrast
to the case of noncooperative investment, we ﬁnd a monotone relationship between the R&Dlevel and the weight of relative performance α.
Proposition 2 Suppose that two ﬁrms compete in a product market, and they make their R&D
investments cooperatively. Then, the equilibrium R&D investment level I C is decreasing in αfor −1 < α < 1.
11

Explicit collusion in product markets is usually illegal, while cooperation at the R&D stage is often permitted

by anti-monopoly legislations.

8We explain the intuition behind Proposition 2. A decrease in d (the common cost of twoﬁrms) lowers the price. Since dp/dd = 2/(3 − α), dp/dd is increasing in α. In other words, when
α is large, a cost reduction by R&D reduces the equilibrium price substantially; thus, ﬁrms lose
incentives for R&D. This yields a smaller R&D level as α is larger.
Proposition 2 suggests that under joint implementation of R&D, spiteful preference can be
obstacles to innovations.

5

OligopolyIn this section, we move back to the model with noncooperative investment and discuss anoligopoly version. Regarding the payoﬀ function based on relative performance, there are several
diﬀerent formulations: (a) each ﬁrm cares about the average proﬁts of the rivals (the other ﬁrms);(b) each ﬁrm cares about the highest proﬁt ﬁrms among the rivals; and (c) each ﬁrm has one
speciﬁc rival as a benchmark ﬁrm and cares about its proﬁt only. Since all formulations yieldexactly the same results in our model, we adopt the ﬁrst formulation.
There are n(≥ 3) symmetric ﬁrms and they engage in the two-stage game formulated in
Section 2. According to formulation (a), the payoﬀ of ﬁrm i (i = 1, 2, ...n) is given by:
Ui = π1 −

α ∑
πj .
n−1
i̸=i

Let β := α/(n − 1). If we interpret β as an indicator of the degree of envy or altruism, it isnatural to assume that β ∈ (−1, 1). On the other hand, if we interpret β as an indicator of thedegree of competition, it is natural to assume that β ∈ (−1, 1/(n − 1)), because the case whereβ = 1/(n − 1) corresponds to the perfect competition.
Consider the second stage competition (quantity competition). The ﬁrst-order condition ofeach ﬁrm is as follows:a − 2yi − (1 − β)

∑
j̸=i

9

yj = ci .

(7)

The reaction function of ﬁrm i is as follows:yi = Ri (Y−i : β) =where Y−i =a − ci − (1 − β)Y−i
,
2

(8)

∑

j̸=i yj . Summing each side in (7) for all ﬁrms, we havena − 2Y − (1 − β)(n − 1)Y =

∑

ci .

(9)

i

Rearranging (9), the second stage total output is obtained:∑na − i ci
Y =
,
2+B

(10)

where
B := (1 − β)(n − 1).
Note that Y is increasing in β. From (7) and (10), we have the second stage output of each ﬁrm:∑
(1 + β)a − (1 + β + B)ci + (1 − β) j̸=i cjEyi =.
(1 + β)(2 + B)
The equilibrium proﬁt of ﬁrm i given the ﬁrst stage actions of ﬁrms is denoted by
]
}[
{
∑
∑
a[1 − (n − 1)β] − (1 + B)ci + j̸=i cj (1 + β)a − (1 + β + B)ci + (1 − β) j̸=i cj
E
πi =
−Ii .
(1 + β)(2 + B)2
Consider the ﬁrst stage competition. Each ﬁrm i maximizes Ui = πiE − β∑Ej̸=i πjwith

respect to Ii . We again restrict our attention to the symmetric equilibrium. Diﬀerentiating Ui
with Ii , and then substituting cj = c for all ﬁrms we obtain−J(β, n)c′ = 1,
whereJ(β, n) :=(a − c)(β + β 2 + 2n − 2β 2 n − βn2 + β 2 n2 ).
(2 + B)2

Then, we have the following result.

10

Proposition 3 Suppose that n(≥ 3) ﬁrms compete in a market, and they make their R&D
investments independently. Then, the equilibrium R&D investment level I E is decreasing in β
for β < β̄ and is increasing in β for β > β̄, where β̄ := (n − 1)/(n + 1).
The threshold value of β in Proposition 3 is increasing in n. This indicates that an increasein β (and so α) is more likely to decrease R&D investments when n is large. As we discussed
in Section 3, an increase in α has two countervailing eﬀects. On the one hand, an increase in
α increases the equilibrium output of each ﬁrm and stimulates R&D. On the other hand, anincrease in α reduces |Ri′ | and so reduces the strategic value of R&D. The former eﬀect becomes
weaker when n is large, and thus, the latter eﬀect more likely dominates the former when n islarge.

6

Welfare Analysis

In this section, we examine welfare implications in duopoly with non-cooperative investment.
The equilibrium investment level is compared to the second-best investment level. Social welfare
W is the consumer surplus plus proﬁt of ﬁrms:
∫ Y
W :=

p(y)dy − p(Y )Y + π1 + π2 .

0Suppose that the social planner cannot control the competition in the second stage and cancontrol only I, the R&D investment level of each ﬁrm (the second-best problem). This problem
is intensively discussed in the literature of R&D competition. Let I ∗ denote the second-best
R&D level.12 We compare I ∗ with I E , the equilibrium R&D investment level.
The ﬁrst-order condition for the social planner is given by[ 4a + 2aα − 2aα2 − 11c + 5αc + 7c − 7αc + 2α2 c ]
dW E
1
1
2
2
2 ′
=−
c1 − 1 = 0.
dI1
(3 − α)2 (1 + α)
12

(11)

We implicitly assume that the second-best outcome is symmetric. This holds true for suﬃciently large c′′i .

11

Since c′′i is positive and suﬃciently large, the second-order condition is satisﬁed. Substituting
c1 = c2 = c into (11), we have that −K(α)c′ = 1 must be satisﬁed, whereK(α) :=2(a − c)(2 − α).
(3 − α)2

A larger K implies a higher level of I ∗ , and I ∗ > (<)I E if and only if K(α) > (<)G(α) (thedeﬁnition of G(α) is given in Section 3).
We now state welfare implications.
Proposition 4 Suppose that two ﬁrms compete in a market, and they make their R&D investments independently. Then,(i) I ∗ is increasing in α for −1 < α < 1.
(ii) I ∗ < I E for −1 < α < 0, and I ∗ > I E for 0 < α < 1.
Proposition 4(i) states that the second-best R&D level is increasing in the reciprocal parameter α. This result is very intuitive. The larger the output each ﬁrm yields, the greater is thesocial beneﬁt of R&D; further a larger α yields the larger output.
Proposition 4(ii) states that when α is negative (the market is collusive), the equilibrium
R&D is excessive, while a positive α yields insuﬃcient R&D. We explain the intuition. When
α is negative, the output level is low. Thus, the social gain of R&D is small, while ﬁrms makerelatively large investments in R&D (see Proposition 1). This yields excessive investment. The
social optimal level of R&D is increasing in α and its equilibrium level is decreasing in α forα ≤ 1/3, I E − I ∗ is decreasing in α; it happens to be zero when α = 0. An increase in α fromzero yields the insuﬃcient investment (I E < I ∗ ). See Figure 1.
[Figure 1 AROUND HERE]

12

7Concluding Remarks

This paper develops a relative performance approach in the context of R&D competition; weinvestigate the two-stage R&D game in oligopoly markets where ﬁrms’ payoﬀs depend on bothabsolute and relative proﬁts. Firm i’s payoﬀ is πi − απj , where πi is its own proﬁt, πj is therival’s proﬁt, and α ∈ (−1, 1). We ﬁnd the U-shaped relationship between α and the levels of
R&D investments when the ﬁrms choose the levels of R&D investments independently. This
result indicates that innovation is quite active in both altruistic and spiteful preferences, while
it is less active in the intermediate situations. We also show that the second-best R&D level isincreasing in α, and it is more likely to exceed the equilibrium R&D level when α is large.
The relative performance approach smoothly connects industrial organizations and preferences of ﬁrms. Our approach contains Bertrand and Cournot competition as special cases, anda larger α indicates tougher competition in product markets. Therefore, this approach connects
the standard Cournot situation with the standard Bertrand situation in a peculiar way. Moreover, it has the potential to reveal important properties between toughness of competition andmarket performance that have thus far been overlooked. To investigate a general property of therelationship between market competition and performance remains an issue for future research.

13

Appendix: Proofs
Proof of Proposition 1: Diﬀerentiating G(α) with respect to α, we havedG(α)(a − c)(3α − 1)=.
dα(3 − α)3(12)(12) is negative for α < 1/3 and positive for α > 1/3. This implies that given c1 = c2 = c, themarginal beneﬁt of R&D is decreasing (increasing) in α for α < (>) 1/3. This yields Proposition
1.

Q.E.D.

Proof of Proposition 2: Diﬀerentiating H(α) with respect to α, we havedH(α)
2(a − d)(1 − α)=−.
dα(3 − α)3(13) is negative for −1 < α < 1.

(13)
Q.E.D.

Proof of Proposition 3: We prove this proposition along the same lines as Proposition 1.
Diﬀerentiating J(β, n) with respect to β, we have∂J(β, n)(a − c)(n − 1)2 (1 + β − n + βn)=.
∂β[2 + (1 − β)(n − 1)]3

(14)

Note that (a−c) and 2+(1−β)(n−1) is always positive. Hence, (14) is negative for 1+β−n+βn <0 and positive for 1 + β − n + βn > 0. A simple calculation leads us to∂J(β, n)n−1∂J(β, n)n−1
< 0 for β <and> 0 for β >
.
∂β
n+1
∂β
n+1
This implies that given cj = c for all j, the marginal beneﬁt of R&D is decreasing (increasing)in β for β < (>) (n − 1)/(n + 1). This yields Proposition 3.

14

Q.E.D.

Proof of Proposition 4: (i) Diﬀerentiating K(α) with respect to β, we havedK(α)
2(1 − α)(a − c)=.
dα(3 − α)3This implies that dK/dα > 0 for all α ∈ (−1, 1). Thus, we have Proposition 4(i).
(ii) Note that I ∗ > (<)I E if and only if K(α) > (<)G(α). The simple calculation leads us toK(α) − G(α) =(a − c)(1 − α)α.
(3 − α)2This implies Proposition 4(ii).

Q.E.D.

15

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I

IE

I∗

α
−1
+1
Figure 1: Welfare Implication (n = 2)19