ESADE WORKING PAPER Nº 265 May 2017

Asymmetries, Passive Partial Ownership Holdings, and Product Innovation

Anna Bayona

Àngel L. López

Supply Function Competition, Private Information, and Market Power: A Laboratory Study

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Asymmetries, Passive Partial Ownership Holdings, and

Product Innovation

Anna Bayona � Ángel L. López y

May 2017

Abstract

We study how asymmetries a¤ect R&D investments, competition, and welfare in markets

with passive partial ownership holdings between rival �rms. The asymmetries that we

consider can arise either because one �rm enjoys an initial competitive advantage, or because

the passive partial ownership holdings between rivals are unequal. In contrast to previous

�ndings, we �nd that, due to asymmetries, passive partial ownership holdings can increase

total surplus in markets with competition in prices and quality-enhancing R&D with no

spillovers. In these markets, our �nding suggests that competition authorities should take

into account the potential bene�cial e¤ects of asymmetric passive partial ownership holdings.

Keywords: partial ownership, minority shareholdings, R&D investments, price competi-

tion, competition policy

JEL codes: D43, L11, L40, G34

1 Introduction

There has been a recent surge in passive partial ownership holdings (hereafter PPO) between

rival �rms which have attracted the attention of competition authorities around the world.1 An

important characteristic of these are the asymmetries between �rms, either because one �rm

enjoys an initial competitive advantage, or because the PPO holdings between rivals are unequal.

Our objective is to study, in the presence of PPO, how asymmetries between rivals a¤ect R&D

investments, competition in the product market, and welfare. In contrast to previous �ndings,

we �nd that, due to asymmetries, PPO can increase total surplus in markets with competition

in prices and quality-enhancing R&D with no spillovers.

We consider a two-period model with two �rms. In the �rst period, �rms invest in R&D to

increase product quality. We examine markets where R&D spillovers are negligible due to high

�Bayona: ESADE Business School. E-mail address: anna.bayona@esade.eduyLópez: Departament d�Economia Aplicada, Universitat Autònoma de Barcelona, and Public-Private Sector

Research Center, IESE Business School. E-mail address: angelluis.lopez@uab.cat1These are subject to merger control in the UK, Germany and the US, among others. The European Com-

mission (EC, 2014, 2016) is currently reviewing the issue, speci�cally in cases where minority shareholdingsacquisitions generate a "competitive signi�cant link", which refers to a situation where either of the followingconditions is satis�ed: a) acquisitions of minority shareholdings in a competitor or vertically related company;b) when either the level of acquired shareholding is around 20% or is greater than 5% but it is accompanied withadditional rights.

1

patent protection. In the second period, �rms compete in prices à la Hotelling for the demand of

a di¤erentiated product that is purchased by a continuum of consumers. We study how di¤erent

ownership structures a¤ect equilibrium outcomes, and focus on those in which �rms own (small)

PPO holdings of rivals with no control rights. Our paper examines how two distinct types of

asymmetries a¤ect incentives to invest in R&D, and as a result, �rms�competitive position in

the second period. One asymmetry concerns unbalanced �nancial interests by rivals. We model

this asymmetry by supposing that one �rm (the acquirer) has a PPO stake in the rival, while

the rival (or target �rm) does now have a PPO holding in the �rm. The other asymmetry is

related to situations in which one of the �rms enjoys a larger initial competitive advantage since

because of brand loyalty or a higher-quality product.

We �nd that, in markets with symmetric PPO holdings and no competitive advantage by

neither �rm, an increase in PPO holdings rise equilibrium prices and reduce the total R&D

investment. In the �rst period, each �rm is aware that an increase in its price increases the

rival�s pro�t, which can then be partially internalized by the �rm since it owns a fraction of the

�rm�s equity. In addition, because investing in R&D decreases the rival�s pro�t, and it is costly,

�rms under-invest in R&D in relation to in markets with no PPO.

We then consider markets with symmetric PPO holdings and in which one �rm has an initial

competitive advantage. We show that the �rm with a competitive disadvantage invests less in

R&D with symmetric PPO holdings than with no PPO holdings. This is because increasing

R&D is less e¤ective at generating pro�ts, and yet because of symmetric PPO, the �rm with

disadvantage will be able to appropriate a share of the rival�s pro�ts. The �rm with a competitive

disadvantage will sell to less than a half of the market. If the competitive advantage is su¢ ciently

large, the �rm with a competitive advantage may invest more in R&D with symmetric PPO

than with no PPO since the bene�t of further increasing its market share outweights the cost

of investing in R&D.

With asymmetric PPO holdings and neither �rm enjoying an initial competitive advantage,

we �nd that the target �rm invests more in R&D than the equivalent �rm in a market with

no PPO holdings, while the acquirer invests less. The target �rm has to over-invest in R&D

in order to generate a positive R&D di¤erential which leads to a higher market share, and

potentially higher prices, thus augmenting revenues, and pro�ts. By contrast, the acquirer has

an incentive to reduce the investment in R&D as asymmetric PPO holdings increase since it

appropriates a higher proportion of the target�s pro�ts generated both through the increase

in the target�s prices and market share. As a result, the acquirer sells to less than half of

the market. Similar results hold when the target �rm enjoys an initial competitive advantage.

However, when the acquirer has a su¢ ciently large competitive advantage, then it can then

happen that its R&D investment increases as asymmetric PPO holdings increase: the acquirer

invests in R&D in order to generate a product of greater quality that will be enjoyed by a larger

proportion of consumers.

Our main �nding is that asymmetries in markets with PPO may lead to a larger total surplus

compared to markets with no PPO. In markets where �rms have asymmetric PPO holdings and

neither �rm enjoys an initial competitive advantage, as PPO holdings increase, the target �rm

increases its R&D investment but also its market share. This means that aggregate utility

2

increases since more consumers buy the good of higher product quality generated by the R&D

investment in relation to markets with no PPO holdings. Total surplus with asymmetric PPO

will be larger than with no PPO if the marginal utility due to R&D is su¢ ciently large in relation

to cost of investing in R&D and to the aggregate transport cost. Nevertheless, consumer surplus

is lower with PPO compared to no PPO, while producer surplus is higher with PPO compared

to no PPO.

In contrast, total surplus with symmetric PPO and neither �rm enjoying a competitive

advantage is always lower than in markets with no PPO. This is because �rms under-invest in

R&D, which implies that a larger proportion of consumers buy a good of lower quality. This

decreases the aggregate utility more than the total cost savings due to a lower investment in

R&D. However, if one of the �rms enjoys a su¢ ciently large initial competitive advantage then

total surplus with symmetric PPO is also larger than total surplus with no PPO. The �rm

with the initial competitive advantage invests more in R&D than the �rm with a disadvantage,

thus generating a positive R&D di¤erential, which leads to a higher market share for the �rm

with the initial competitive advantage. As a result, aggregate utility is larger than in markets

with no PPO due to: (1) higher proportion of consumers buying the good with a greater initial

gross utility; (2) higher proportion of consumers buying the good of higher quality due to the

larger investment in R&D by the �rm with the initial competitive advantage. Total surplus

with symmetric PPO and an asymmetry in competitive advantage is larger than with no PPO

when this increase in aggregate utility outweights transport and R&D investment costs.

Total surplus may also be larger with PPO than with no PPO when the two types of

asymmetries are present. If the target �rm enjoys an initial competitive advantage then both

asymmetries reinforce each other and the target �rm over-invests in R&D. Hence, total surplus

is larger with asymmetric PPO than with no PPO if the marginal utility of investing in R&D

is large in relation to transport and R&D investment costs. If the acquirer enjoys an initial

competitive advantage then it is also possible that total surplus is larger with PPO than with no

PPO since the competitive advantage mechanism will mean that it might invest more in R&D

compared to no PPO. However, it is less likely that total surplus is greater with PPO than with

no PPO in relation to when the target has a competitive advantage.

Our paper is related to three branches of literature. First, it is related to the literature on co-

operative R&D and spillovers, which among others includes the seminar articles of d�Aspremont

and Jacquemin (1988), Suzumura (1992), Ziss (1994), Kamien et al. (1992), and Leahy and

Neary (1997). This literature focuses on cost-reducing R&D investments, while we consider

quality-enhancing R&D investments. The second branch of literature examines the e¤ects of

passive partial ownership holdings on competition. The seminar articles of Bresnahan and Salop

(1986), Reynolds and Snapp (1986), and Allen and Phillips (2000) show that PPO holdings are

anti-competitive and result in higher prices.2 However, in these articles R&D investments are

absent. Finally, our paper is related to López and Vives (2016) which consider cost-reducing

R&D investment with spillovers, and show that PPO holdings may increase welfare only if mar-

kets are not too concentrated, demand is not too convex and moreover spillovers are su¢ ciently

large. In contrast, in the presence of quality-enhancing R&D investments, we show that total

2These results have also been corroborated by the empirical literature, such as in Azar et al (2015).

3

welfare may increase with PPO holdings even if there are no spillovers in the industry.

The paper is organized as follows. Section 2 describes the model. Section 3 �nds the equi-

librium prices, market shares and R&D investments for di¤erent ownership structures. Section

4 conducts welfare analysis. Section 5 discusses the strategic incentives to acquire minority

shareholdings. Section 6 concludes.

2 Model

There are two �rms, i; j = A;B with i 6= j, and !i is �rm i�s share of pro�ts in �rm j. The

pro�t of �rm i�s portfolio of investments is

�i = (1� !j)�i + !i�j ; (1)

where

�i = pisi ��x2i2; (2)

where si is �rm i�s market share. We are interested in the case where there are asymmetries

between the equity interests of the two �rms: !A = !; !B = 0. In order to analyze this question,

we consider the benchmark of no PPO (!A = !B = 0), and study the case of symmetric PPO,

where PPO holdings between rivals are equal (!A = !B = !).3 We restrict that PPO satis�es

0 � !i; !j < 1=4. 4 For a given ownership structure (!A;!B); we de�ne the total equilibriumR&D investment as X�(!A;!B) = x

�A(!A;!B) + x

�B(!A;!B).

There is a mass of consumers uniformly distributed along the unit line that have a unit

demand. Consumers can purchase the good either from �rm A or �rm B, and we assume full

participation. We use Hotelling model for this type of demand.

The timing of the model is as follows.

� At t = 1, each �rm chooses an R&D investment, xi, by maximizing the pro�ts from its

portfolio of investments, �i.

� At t = 2, given an R&D investment, each �rm sets a price, pi, for a di¤erentiated product,and subsequently consumers purchase from either �rm. A consumer that buys from �rm

i obtains the following utility

Ui(xi0; xi)� tjqi � qj � pi; (3)

where t > 0 is the product di¤erentiation parameter; qi�f0; 1g is the location of the �rm(without loss of generality assume that qA = 0; qB = 1); and q�[0; 1] is the location of a consumer.

3Passive partial ownership (PPO) stakes are also often called passive minority �nancial interests, non-controlling minority shareholdings, or passive cross-ownership.

4There is not a commonly agreed thresdhold about what consistute non-controlling minority shareholdingsby competitors. However, competition authorities often inspet the non-controlling minority shareholdings bycompetitors that are between 15% and 25% (Salop and O�Brien 2000).

4

Hence, a consumer that is located at q loses tjqi� qj units of utility when he purchases the goodlocated qi. The utility function of a consumer that buys from �rm i is as follows

Ui(xi0; xi) = x

i0 + �xi; (4)

where xi0 is the initial gross utility of buying good sold by �rm i; consumer�s utility function

increases by � > 0 for each unit of R&D invested by �rm i. Hence, R&D is of the quality-

enhancing type.

Let q be the consumer that is indi¤erent between buying A and B, then:

UA(xA0 ; xi)� tj � qj � pA = UB(xB0 ; xi)� tj1� qj � pB:

Therefore, the market share of �rm A is:

sA =1

2t((xA0 � xB0 ) + �(xA � xB) + (pB � pA) + t); (5)

and �rm B�s market hare is sB = 1 � sA. Without loss of generality, denote sA = s and

sB = 1 � s. De�ne the di¤erence in the initial gross utility from buying from �rm A and �rm

B as: �A � xA0 � xB0 , and conversely, �B = xB0 � xA0 = ��A.As explained in the introduction, we consider two distinct types of asymmetries.

� Asymmetry due to �rm i enjoying an initial competitive advantage (disadvantage). In ourmodel this translates to �i > 0 ( �i < 0), which leads to a positive (negative) di¤erence

in initial gross utility derived from buying from �rm i.

� Asymmetry due to unequal PPO holdings between �rms. We model this asymmetry as

!i = ! and !j = 0, for i 6= j.

The equilibrium concept used is the Subgame Perfect Equilibrium. A �rm�s strategy con-

sists of an investment in R&D and a subsequent pricing strategy which is based on the R&D

investment. In terms of welfare, we consider the second-best welfare benchmark such that the

planner chooses the R&D that maximizes total surplus, taking as given the equilibrium prices

set by the two �rms in the second stage. We also compare total surplus at the equilibrium

allocation for di¤erent ownership structures.

3 Equilibrium prices, market shares and R&D Investments

3.1 Benchmark models: No PPO and symmetric PPO

We study two benchmark models with respect to the ownership structure: markets with sym-

metric PPO (!A = !B = !), and markets with no PPO (! = 0). We also allow asymmetries in

initial competitive advantage to be present, i.e. �i 6= 0.We �rst �nd the equilibrium of the model for symmetric PPO. Pro�ts of �rms i; j = A;B

with i 6= j are:

5

�i(!;!) = (1� !)�i + !�j ; (6)

where �i = pisi � �x2i2 , and si =

12t(�i + �(xi � xj) + (pj � pi) + t). We solve by backward

induction and �nd the optimal prices at t = 2. The �rst order condition for each �rm i satis�es@�i@pi

= 0. Hence, the best response price equations for �rms i; j = A;B are such that for i 6= j:

pi(!;!) =(1� !)�i + (1� !)�(xi � xj) + pj + (1� !)t

2(1� !) :

For given R&D investments, we observe that the price best-reply function is upward sloping

(i.e. competition at t = 2 is of the strategic complements type). Solving for the �xed point, we

obtain the prices as functions of R&D investments at t = 1:

pi(!;!) = (1� !)�i(1� 2!) + t(3� 2!) + �(xi � xj)(1� 2!)

(2! � 3) (2! � 1) :

Substituting for these optimal prices we �nd that the position of the indi¤erent consumer

is: si = 16t (3t+�i + �(xi � xj)). We now solve for the equilibrium R&D investments at t = 1.

The �rst order condition for �rm i satis�es: @�i@xi= 0. The second order condition requires that

�t (2! � 3)2 � �2 > 0, and the stability of the equilibrium that �t (2! � 3)2 � 2�2 > 0. As a

result, we obtain the best response functions for R&D of each �rm:

xi = �

��i � t (1� !) (2! � 3)(�t (2! � 3)2 � �2)

� �

(�t (2! � 3)2 � �2)xj

�:

We note that R&D investment at t = 1 is of strategic substitutes type.

The strategic e¤ect can be shown to be negative.5 It follows that, at the interior equilibrium,

the direct e¤ect is positive, in which case �rm i under-invests in relation to the direct e¤ect.

Recall that the sign of the strategic e¤ect depends on whether R&D investment makes �rm i

tough or soft, and on whether second period actions are strategic substitutes or complements.

R&D investment makes �rm i tough since@�j(xi;p

�i ;p

�j )

@xi< 0. Furthermore, second period prices

are strategic complements. As a result, the strategic e¤ect is necessarily negative. That is,

investment in R&D makes �rms tough, which combined with strategic complementarity in the

second period yield a negative strategic e¤ect, which results in under-investment in R&D. Firms

under-invest in R&D so as not to trigger an aggressive response from the rival �rm, also known

as a "puppy dog" strategy (Fudenberg and Tirole, 1984).

We now solve for the �xed point. The following proposition states the equilibrium outcomes

with symmetric PPO.

PROPOSITION 1 If the second order and stability conditions are satis�ed, i.e. �t (2! � 3)2��2 > 0, for each �rm i, at equilibrium

x�i (!;!) =�

� (3� 2!)

���i(3� 2!)

t� (2! � 3)2 � 2�2+ (1� !)

�; (7)

5The following are satis�ed: @�i@pj

dp�jdxi

< 0; @�i@xi

> 0

6

with X�(!;!) = x�A(!;!) + x�B(!;!) =

2�(1�!)�(3�2!) ,

p�i (!;!) =(1� !)

(3� 2!) (1� 2!)

�t(3� 2!) + �i(1� 2!)

�1 +

2�2

(t� (2! � 3)2 � 2�2)

��; (8)

and

s�i (!;!) =1

2

�1 +

��i(3� 2!)t� (2! � 3)2 � 2�2

�: (9)

The special case of no PPO can be obtained from Proposition 1 by setting ! = 0.

COROLLARY 1 (no PPO). If the second order and stability conditions are satis�ed, i.e. 9�t��2 > 0, for each �rm i, at equilibrium

x�i (0; 0) =�

3�

�1 +

3��i(9t�� 2�2)

�;

and X�(0; 0) = x�A(0; 0) + x�B(0; 0) =

2�3� ,

p�i (0; 0) = t

�1 +

3��i9t�� 2�2

�; (10)

and

s�i (0; 0) =1

2

�1 +

3��i9t�� 2�2

�: (11)

In markets with symmetric PPO and no competitive advantage by neither �rm, equilibrium

prices are higher than with no PPO since competition is softer in the second stage, and as a

result �rms invest less in R&D than if �rms had no �nancial links. The reasoning is as follows.

With symmetric PPO, �rm i at t = 1 takes into account that at t = 2 an increase in pi increases

�rm j�s pro�t, which can be partially internalized by �rm i since it owns a fraction ! of �rm j.

In addition, due to symmetry and no initial competitive advantage, each �rm will still sell to

half of the market (since there is full participation). Because investing in R&D decreases the

rival�s pro�t, and it is costly, �rms under-invest in R&D in relation to in markets with no PPO.

For symmetric PPO, Table 1 describes the comparisons of equilibrium outcomes with sym-

metric PPO in relation to markets with no PPO for various signs of �i.

Table 1. Symmetric PPO: Signs of the comparisons.

sign(:::) �i = 0 �i < 0 �i > 0

x�i (!;!)� x�i (0; 0) � � sign

��i �

(9t��2�2)(t�(2!�3)2�2�2)12t�2(3�2!)(3�!)

�X�(!;!)�X�(0; 0) � � �

p�i (!;!)� p�i (0; 0) + sign

�(9t��2�2)(t�(2!�3)2�2�2)

(1�2!)� ��i�3t�(3� 2!)� 2�2(5� 2!)

��s�i (!;!)� s�i (0; 0) 0 � +

Suppose that �rm i has an initial competitive disadvantage, i.e. �i < 0. With symmetric

PPO, for given levels of R&D and prices, the market share of �rm j will be larger than the

7

market share of �rm i. More consumers will �nd it optimal to purchase from �rm j because a

larger proportion derive a greater initial gross utility by purchasing from j. As a result, �rm

j will be able to charge higher prices than �rm i, and generally the equilibrium prices of both

�rms increase with ! because of softer competition. Even when ! = 0, �rm i has a lower level

of R&D investment than �rm j. In addition, �rm i invests less in R&D with symmetric PPO

compared to no PPO since it is more costly for this �rm to gain market share compared to �rm

j. With symmetric PPO, �rm i can take advantage of the R&D investment by �rm j, which

is more e¤ective at generating pro�ts. The combined e¤ect is that �rm i has a lower market

share than �rm j.

Suppose now that �rm i has a competitive advantage, i.e. �i > 0. The argument is the

opposite to the one just described in the previous paragraph, except from the sign of how R&D

investment with PPO holdings compares to no PPO holdings since it depends on the size of the

competitive advantage. If �i is large then �rm i may invest more in R&D than when there are

no PPO holdings. This is because the positive R&D di¤erential between �rms lead to a larger

di¤erential in prices at t = 2, which translate in a larger market share for �rm i, and higher

corresponding pro�ts. In contrast, if �i is su¢ ciently small, then the cost of increasing R&D

outweights the bene�t derived from the potential increase in revenues. Hence, if�i is su¢ ciently

small then �rm i invests less in R&D with symmetric PPO than with no PPO holdings.

3.2 Asymmetric PPO

We consider the case when PPO holdings between rivals are unbalanced, and model it by

assuming that !A = !; !B = 0. This formulation captures the situation in which the acquirer

(A) has a PPO holding of the target �rm (B), while the target �rm does not have any PPO

holdings of the acquirer. Pro�ts of �rms A and B, are

�A(!; 0) = �A + !�B = pAs��x2A2+ !pB(1� s)�

!�x2B2

; (12)

�B(!; 0) = (1� !)�B = (1� !)�pB(1� s)�

�x2B2

�: (13)

Next Proposition states the equilibrium with asymmetric PPO holdings.

PROPOSITION 2 If the second order and stability conditions are satis�ed, i.e. �t(3 � !)2 ��2 > 0, at equilibrium

x�A(!; 0) = �

0@��A(3� !)� �2(2� !) + t� (3� !) �!2 � 5! + 3�� (3� !)

�t� (3� !)2 � 2�2

�1A ; (14)

x�B(!; 0) = �

�� (3� !) (3t��A)� �2 (2� !)� (3� !) (t� (3� !)2 � 2�2)

�; (15)

8

and X�(!; 0) = x�A(!; 0) + x�B(!; 0) =

�(2�!)�(3�!) ,

p�A(!; 0) = t

0@��A(3� !)(1� !) + �2(!2 � 3! � 2) + (3� !)(3 + !)�t�t� (3� !)2 � 2�2

�1A ; (16)

p�B(!; 0) = t

0@�(3� !)��A � (2� !)�2 + 3t�(3� !)�t� (3� !)2 � 2�2

�1A : (17)

and

s�A(!; 0) =1

2

0@��A(3� !)� (2 + !)�2 � t� (2! � 3) (3� !)�t� (3� !)2 � 2�2

�1A ; (18)

while s�B(!; 0) = 1� s�A(!; 0).

Suppose that neither the target (B) nor the acquirer (A) enjoy an initial competitive advan-

tage. In the second stage, for given R&D investments, the acquirer takes into account that it

internalizes a share ! of the rival�s pro�t, while the target �rm only obtains a share 1�! of itsown pro�ts. For given R&D investments, an increase in pA by one unit triggers an increase in

pB by 12 , while an increase in pB by one unit triggers an increase in pA by

1+!2 . Because of the

asymmetry in PPO holdings, prices at t = 2 will be more moderate in relation to the symmetric

PPO. At t = 1, the target �rm augments R&D as PPO holdings increase as a means of generate

a positive R&D di¤erential which leads to a higher market share, and potentially higher prices,

thus increasing revenues and pro�ts. By contrast, the acquirer has an incentive to reduce the

investment in R&D as ! increases since it appropriates a proportion ! of the surplus generated

by target �rm both through the increase in prices and market share. In addition, an increase

in the R&D investment by the acquirer decreases the target�s pro�ts.

Table 2 shows the comparison of equilibrium outcomes with asymmetric PPO with respect

to the no PPO benchmark.

Table 2. Asymmetric PPO: Signs of the comparisons.

sign(:::) �i = 0 �A < 0;�B > 0 �A > 0;�B < 0

x�A(!; 0)� x�i (0; 0) � � sign

��A �

((3�!)(9�2!)t���2)(9t��2�2)3t�2(3�!)(6�!)

�x�B(!; 0)� x�i (0; 0) + + sign

�(9t��2�2)(t�(3�!)(6�!)+�2)

3t�2(!�3)(!�6) ��A�

X�(!; 0)�X�(0; 0) � � �p�A(!; 0)� p�i (0; 0) sign

�2t�� �2

sign

n�2t�� �2

�(3� !)� 2��A 3t�(3�!)��

2(4�!)(9t��2�2)

op�B(!; 0)� p�i (0; 0) + + +

s�A(!; 0)� s�i (0; 0) � � signn�A � ! �

2+t�(3�!)�(3�!)

os�B(!; 0)� s�i (0; 0) + + �sign

n�A � ! �

2+t�(3�!)�(3�!)

oTable 2 shows that with asymmetric PPO holdings and �A � 0, the target �rms invests

more in R&D compared to an equivalent �rm in a market with no PPO holdings, while the

9

acquirer invests less. In fact, we can show that if �A = 0 then the following relationship holds:

x�B(!; 0) > x�i (0; 0) > x

�i (!;!) > x

�A(!; 0). However, when the acquirer has a su¢ ciently large

competitive advantage, i.e. �A > 0, it can then happen that the acquirer increases its R&D

investment as asymmetric PPO holdings increase. This is because the acquirer starts with an

advantage in market share and quality, and as a result, it can charge higher prices. Because

the target �rm has a small market share it is more e¤ective that the acquirer invests in R&D

in order to generate a product of greater quality that will be enjoyed by a larger proportion

of consumers. Overall, the sum of R&D investments by both �rms is smaller with asymmetric

PPO holdings than with no PPO holdings.

We also observe that the price of the target �rm is always larger with asymmetric PPO

holdings than with no PPO holdings irrespective of the sign of �A. However, the equilibrium

price of the acquirer may be larger or smaller than an equivalent �rm with no PPO holdings. If

�A = 0 and �2 is large in relation to t� then consumers give a high value to quality and generally

prefer to buy the good produced by the target �rm since it invests more in R&D, and hence

its product is of higher quality. The acquirer then reduces prices in order to be competitive,

and not further reduce its market share. In fact we can show that sign (p�A(!; 0)� p�B(!; 0)) =sign

�t� (3� !)� �2 (4� !)

�. Consequently, if �A � 0 then the market share of the target

(acquirer) �rm will be larger (smaller) with asymmetric PPO holdings than with no PPO

holdings. If the acquirer enjoys a su¢ ciently large competitive advantage (i.e. if �A is positive

and su¢ ciently large) then market share of the acquirer will be larger for some ! > 0 in relation

to when ! = 0, until the target �rm�s proposal is more attractive.

For an illustration of the comparison between equilibrium outcomes for di¤erent ownership

structures, see Figures 1 and 2.

4 Welfare analysis

For a given ownership structure (!A;!B), we de�ne producer surplus, consumer surplus and

total surplus. Producer surplus is

PS(!A;!B) = �A(!A;!B) + �B(!A;!B) = pAs+ pB(1� s)��

2(x2A + x

2B); (19)

where sA = s = 12t((x

A0 � xB0 ) + �(xA � xB) + (pB � pA) + t). We notice that producer surplus

is equal to the total revenue minus the total cost of investing in R&D.

Consumer surplus is equal to

CS(!A;!B) =

sZ0

(xA0 + �xA � ty � pA)dy +1Zs

(xB0 + �xB � t(1� y)� pB)dy; (20)

which is the sum of consumers�utility from buying either of the �rms minus the total payment

transferred to each of the �rms and minus the total costs due to product di¤erentiation.

Total surplus is TS(!A;!B) = CS(!A;!B) + PS(!A;!B), and thus equals

10

Figure 1: Equilibrium outcomes with respect to PPO.

Figure 2: Equilibrium outcomes with respect to output.

11

TS(!A;!B) = (xA0 s+ x

B0 (1� s)) + �(xAs+ xB(1� s))�

t

2(s2 + (1� s)2)� �

2(x2A + x

2B): (21)

The next Proposition compares total surplus at the equilibrium allocation, TS�(!A;!B),

for di¤erent ownership structures.

PROPOSITION 3 Suppose that �i = 0. It always holds that

TS�(0; 0) > TS�(!;!):

If � is su¢ ciently large and t� su¢ ciently small then

TS�(!; 0) > TS�(0; 0) > TS�(!;!);

while if � is su¢ ciently small and t� su¢ ciently large then

TS�(0; 0) > TS�(!;!) > TS�(!; 0)

for some ! > 0.

The Proposition shows that in markets with no asymmetries due to an initial competitive

advantage (�i = 0), total surplus with symmetric PPO is always lower than in markets without

PPO. As PPO increases both �rms steeply increases prices and under-invest in R&D in relation

to markets with no PPO. Yet both �rms serve half of the market. Due to PPO, consumer

surplus always decreases more than the increase in producer surplus. This is because the under-

investment in R&D causes a larger decrease in aggregate utility (since a larger proportion of

consumers buy a good of lower quality) which is not compensated by the cost-savings of both

�rms due to a lower investment in R&D.

The surprising �nding is that total surplus with asymmetric PPO may be larger than total

surplus with no PPO if the marginal utility due to R&D (�) is su¢ ciently large in relation to

product of the R&D cost parameter, �, and the transportation cost associated with product

di¤erentiation, t. This is because, as PPO holdings increase, the target �rm increases its R&D

investment but also its market share. This leads to a substantial increase in aggregate utility

due to more consumers buying a good of higher quality, generated by the R&D investment,

in relation to when there are no PPO holdings. When � is su¢ ciently large in relation to t�,

this bene�cial allocation e¤ect outweights the increase in costs due to R&D and transport, and

hence total surplus is larger with asymmetric PPO than in markets with no PPO. However,

we �nd that due to PPO holdings, consumers are always worse o¤ while producers are always

better o¤. This is due to the e¤ects of prices on both consumer and producer surplus, which is

eliminated in total surplus since it is just a transfer from consumers to producers. In contrast,

when � is su¢ ciently small in relation to t�, for some large ! total surplus with asymmetric

PPO is even smaller than with symmetric PPO since the joint costs of the over-investment in

R&D and transport do not compensate the positive increase in aggregate utility.

12

The next proposition studies the comparison of total surplus with symmetric PPO and no

PPO when one of the �rms enjoys an initial competitive advantage.

PROPOSITION 4 If �i 6= 0 then

sign fTS�(!;!)� TS�(0; 0)g = sign��2i�� �

;

where

� ��4t��2

�4!3 + 12!2 � 54! + 27

�+ 9t2�2 (6� 5!) (2! � 3)2 � 4�4

�4!2 � 13! + 12

��>

0 for the range of ! considered (i.e. 0 < ! � 14), and � �

�2(3�!)(9t��2�2)2(t�(2!�3)2�2�2)2

9t�(2!�3)2 > 0.

If either of the �rms has a su¢ ciently large initial competitive advantage, i.e. �2i >�� ,

then total surplus with symmetric PPO is larger than with no PPO. Without loss of generality

suppose that �i > 0. Firm i invests more in R&D than �rm j, thus generating a positive R&D

di¤erential, which leads to a higher market share for �rm i. As a result, aggregate utility with

PPO is larger than with no PPO due to: (1) higher proportion of consumers buying the good

with a greater initial gross utility; (2) higher proportion of consumers buying the good of higher

quality generated by R&D investment of �rm i. Total surplus with symmetric PPO is larger

than with no PPO when this increase in aggregate utility outweights the transport and R&D

investment costs.

We have conducted simulations to analyze the comparison of total surplus with symmetric,

asymmetric PPO and with no PPO, which are displayed in the Figure 3.

The previous �gures show the following: (i) When the target �rm enjoys an initial com-

petitive advantage (�B > 0) and � is su¢ ciently large in relation to t� then total surplus

with asymmetric PPO is larger than with no PPO. This is because the mechanism described in

Proposition 4 is reinforced since the target �rm invests more in R&D, and its market share is

greater than if �B = 0. In fact, the level of � that is required for total surplus to be higher with

asymmetric PPO than with no PPO is smaller as �B becomes larger. (ii) When the acquirer

enjoys an initial competitive advantage (�A > 0), the level of � that is required for total surplus

to be higher with asymmetric PPO than with no PPO increases as �A becomes larger. This is

because when ! = 0, R&D investment, price and market share are larger for the acquirer than

the target �rm. As asymmetric PPO holdings increase, the R&D investment, price and market

share of the acquirer decrease, while those for the target �rm increase. For a given level of �,

aggregate utility is lower than if �A = 0 since on average consumers buy a good of a lower

initial gross utility and of a lower quality due to the R&D investment by �rms, which does not

compensate the aggregate transport and R&D costs.

5 Strategic Incentives to Acquire Minority Shareholdings

In the previous sections we have considered exogenous ownership structures. In this section, we

examine the strategic incentives to acquire minority shareholdings.6 Let us consider the case6This issue has been previously analyzed by Flath (1991).

13

Figure 3: Total surplus with symmetric and asymmetric PPO.

of asymmetric shareholdings: �A = �A + !�B and �B = (1 � !)�B, and include a stage 0 inwhich the acquirer, A, decides the PPO holdings to invest in the target, !. If the market is

e¢ cient then �rm A will pay for the equity ! exactly the amount !�B. Therefore, the net pro�t

is �A = �A�!�B = �A, and �rm A will acquire ! from B only if as a result of the acquisition

its operating pro�t (�A) increases. Second-period equilibrium yields p�i (xi; xj), p�j (xi; xj), while

�rst-period equilibrium gives x�i (!) and x�j (!), which replaced into the former expressions gives

p�i (!), p�j (!). Therefore, at equilibrium we can write �i as a function of ! as follows:

�A(p�A(!); p

�B(!); x

�A(!); x

�B(!)).

Note that �A is the operating pro�t, and therefore @�A=@! = 0. We have

d�Ad!

=

0@@�A@pA

dp�Ad!

+@�A@xA

dx�Ad!| {z }1A

DIRECT EFFECT

+

0BB@@�A@pB

dp�Bd!

+@�A@xB

dx�Bd!| {z }

STRATEGIC EFFECT

1CCA � 0,

if d�A=d! < 0, then ! = 0. If the direct e¤ect is negative, then the incentives to acquire

minority shareholdings depend on the sign of the strategic e¤ect.

Consider the �rst-order conditions of the second-period:

@�A@pA

+ !@�B@pA

= 0.

14

Since @�B=@pA > 0, it is clear that @�A=@pA < 0. Clearly, @�A=@pB > 0. Note also that, in

the �rst period for �i(xi; xj ; p�i ; p�j ), the �rst-order condition is

@�A@xA

+ !@�B@xA

= 0,

since the game is tough with respect to R&D investments: @�B=@xA < 0, necessarily at equi-

librium @�A=@xA > 0.7

For t� su¢ ciently large, evaluated at ! = 0, we get

sign

�dp�Ad!

����!=0

�> 0, sign

�dp�Bd!

����!=0

�> 0,

sign

�dx�Ad!

����!=0

�< 0, sign

�dx�Bd!

����!=0

�> 0 and sign

�@�A@xB

����!=0

�< 0.

Therefore, the direct e¤ect is negative (as in the standard game), whereas the strategic e¤ect

has one positive component and one negative component. For t� su¢ ciently large:

d�Ad!

����!=0

=

0B@ (�)@�A@pA

(+)

dp�Ad!

+

(+)

@�A@xA

(�)dx�Ad!| {z }1CA

DIRECT EFFECT

+

0BB@(+)

@�A@pB

(+)

dp�Bd!

+

(�)@�A@xB

(+)

dx�Bd!| {z }

STRATEGIC EFFECT

1CCA � 0.

While the R&D investment choices are strategic substitutes and therefore introduce a nega-

tive strategic e¤ect, price choices are strategic complements and therefore introduce a positive

strategic e¤ect. This positive strategic e¤ect, (@�A=@pB) (dp�B=d!), can make PPO acquisition

pro�table.

The intuition is as follows. At period 2, for some given xi; xj , a higher ! softens the

competitive pressure of �rm A, which results in a higher pA. Since price choices are strategic

complements, �rm B also increases its price: pB. Speci�cally,

@pA@!

= 23t��A � �(xA � xB)

(3� !)2 = 2@pB@!

.

That is, the increase in pA is twice the increase in pB. Since R&D investments make �rm

A tough (@�B=@xA < 0) and prices are strategic complements, at period 1 �rm A adopts a

puppy dog strategy: A under-invests so as not to trigger an aggressive response from its rival.

A higher ! makes the second-period game less tough since prices increase with !, but since

the increase in pB is half of the increase in pA, �rm A under-invests even more to reduce the

competitive pressure in period 2. Thus, xA decreases and because of R&D investments are

strategic substitutes, xB increases. The impact of the changes of the choice variables on A�s

pro�t is intuitively clear:

� The increase in pA decreases �A because A is overpricing (recall that A maximizes �A +!�B)

7 In particular, for �i = ! = 0, @�A=@xA = �=6 > 0.

15

� The decrease in xA decreases �A because A is under-investing (because of the tough natureof the subgame)

� The increase in pB increases �A because for the same price pA, �rm A obtains a higher

market share

� The increase in xB decreases �A because it has a negative direct e¤ect on A�market share,and moreover it makes competition more tough in the second period

Therefore, only the positive strategic e¤ect of ! on rival price can induce A to acquire some

�nancial interests in �rm B.

6 Concluding Remarks

This paper has studied how asymmetries in passive partial ownership (PPO) holdings or asym-

metries in initial competitive advantage a¤ect R&D investments, competition, and welfare. We

build a two-period model with two �rms, such that in the �rst period �rms invest in R&D to

increase product quality, and in the second period �rms compete in prices à la Hotelling for

the demand of a di¤erentiated product that is purchased by continuum of consumers that buy

from either of the �rms. We characterize the Subgame Perfect Equilibrium of the game, and we

consider the second-best welfare benchmark. Our model applies to markets in which rival �rms

have PPO holdings for purely �nancial reasons, and in which PPO holdings are small so that

�rms do not obtain control rights. In addition, our model is relevant in markets where R&D

spillovers are negligible due to high patent protection.

In contrast to previous �ndings, we �nd that PPO can increase total surplus in markets with

competition in prices and quality-enhancing R&D with no spillovers. This is due to asymmetries

in either PPO holdings between rivals, or due to one of the �rms enjoying a su¢ ciently large

initial competitive advantage. The target �rm or the �rm with the initial competitive advantage

over-invest in R&D in relation to when there are no PPO holdings, leading to a higher market

share, which means that a larger proportion of consumers buy a good of higher quality. This

leads to greater aggregate utility than if there were no PPO holdings. If consumers give su¢ cient

high value to quality in relation to cost of R&D and transport then total surplus will be larger.

Our �nding suggests that competition authorities should take into account the potential

bene�cial e¤ects of asymmetric PPO holdings in markets with competition in prices, and quality-

enhancing R&D even if there are no spillovers.

Our paper suggests a few open questions for future research. Future work could study the

welfare e¤ects of asymmetries in active partial ownership stakes. Furthermore, the paper calls

for the development of further empirical work which investigates the role of asymmetries (either

in PPO stakes or in initial competitive advantage) on prices and R&D investments.

16

References

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Industrial Economics, 375-393.

7 Appendix

The following appendix provides a proof of the propositions in the text.

17

Proof of Proposition 1 We start solving backwards and use the �rst order condition at

t = 2: @�i@pi

= 0 for i = A;B. Hence, the reaction functions for for �rms i; j = A;B and i 6= jare

pi(!;!) =(1� !)�i + (1� !)�(xi � xj) + pj + (1� !)t

2(1� !) :

Finding the �xed point we note that

pi(!;!) =(1� !)

(2! � 3) (2! � 1)(�i(1� 2!) + t(3� 2!) + �(xi � xj)(1� 2!));

with si = 12t(�i + �(xi � xj) + (pj � pi) + t) and pj � pi = �2 (�i + �(xi � xj))

1�!3�2! .

Substituting these expressions into pro�ts for each �rm, we can then solve at t = 1 for the

optimal amount of R&D investment: @�i@xi

= 0 for i; j = A;B and i 6= j such that the second

order condition, �t (2! � 3)2 � �2 > 0 is satis�ed and that the stability condition requires that����@2�A@x2A

@2�B@x2B

���� > ���� @2�A@xA@xB

@2�B@xB@xA

���� ;which is equivalent that �t (2! � 3)2 � 2�2 > 0 applies. From the �rst order condition, we

obtain

xi(!;!) =��t (1� !) (2! � 3) + ��i

�t (2! � 3)2 � �2� �2 (1� !)(1� !)(�t (2! � 3)2 � �2)

xi:

Hence, �nding the �xed point, �nd the results stated in Proposition 2.�

Proof of Corollary A Let us draw comparative statics of the equilibrium allocation with

symmetric PPO with respect to !.

@s�(!)

@!=��i

�t� (2! � 3)2 + 2�2

�(t� (2! � 3)2 � 2�2)2

,

and

@p�i (!)

@!= t

(t� (2! � 3)2 � 2�2)2 ��i� (2! � 1)2 (t� (2! � 3)2 � 2�2(5� 4!))(1� 2!)2 (t� (2! � 3)2 � 2�2)2

;

and

@x�i (!)

@!= ��

��9t�� 2�2 � 12t�! + 4t�!2

�2 � 4t�2�i (3� 2!)3�� (2! � 3)2 (�4t�!2 + 12t�! + 2�2 � 9t�)2

:

Let us consider several cases:

� If �i = 0 then @s�(!)@! = 0, @p

�i (!)@! > 0, @x

�i (!)@! < 0.

� If �i < 0 then @s�(!)@! < 0, @x

�i (!)@! < 0, and

sign(@p�i (!)

@!) = sign((t� (2! � 3)2 � 2�2)2 ��i� (2! � 1)2 (t� (2! � 3)2 � 2�2(5� 4!))):

18

� If �i > 0 then @s�(!)@! > 0, and

sign(@x�i (!)

@!) = sign(4t�2�i (3� 2!)3 �

�9t�� 2�2 � 12t�! + 4t�!2

�2);

sign(@p�i (!)

@!) = sign((t� (2! � 3)2 � 2�2)2 ��i� (2! � 1)2 (t� (2! � 3)2 � 2�2(5� 4!))):

�

Proof of Proposition 2 The �rst order conditions for �rm A, @�A@pA= 0, and for �rm B

@�B@pB

= 0 imply

pA(!; 0) =1

2(�A + �(xA � xB) + pB(1 + !) + t):

Substituting the expression of pB into A�s price reaction function we obtain:

pA(!; 0) =1

3� ! (t(3 + !) + (1� !)(�A + �(xA � xB)));

and from the earlier proposition we know that

pB(!; 0) =1

3� ! (3t��A � �(xA � xB));

with s(!; 0) = 12t(3�!)(t(3 � 2!) + �A + �(xA � xB)). Now, we can solve t = 1 optimal

amount of R&D. The �rst order condition for A satis�es: We obtain that @�A@xA

= 0 and the

second order condition is satis�ed if �t(3 � !)2 � �2 > 0, and the stability condition requires

that �t(3� !)2 � 2�2 > 0. The �rst order condition, implies the following reaction function:

xA(!; 0) = ��A + t(!

2 � 5! + 3)� �xB(�t (3� !)2 � �2)

;

and the �rst order condition for B and reaction function are analogous to the benchmark

model without PPO. Hence,

xB(!; 0) =3t�� ��A

(t� (! � 3)2 � �2)� �2xA

(t� (! � 3)2 � �2);

and hence by �nding the �xed point the results of Proposition 3follow.�

Proof of Corollary B Let us draw comparative statics of the equilibrium allocation with

asymmetric PPO with respect to !.

@x�A(!)

@!= ��

�

(9� !) (3� !)3 t2�2 � 2 (3� !)3 t�2�A � (7� 2!) (3� !)2 ��2t+ 2�4

(! � 3)2��t� (3� !)2 + 2�2

�2 ;

19

@x�B(!)

@!= ��

�

2 (3� !)3�At�2 � 6t2�2 (3� !)3 + (3� 2!) (3� !)2 t��2 + 2�4

(3� !)2��t� (3� !)2 + 2�2

�2 ;

@s�A(!)

@!=�3 (! � 3)2 t2�2 + t��2(!2 � 4! � 3) + �A�(2�2 + t� (! � 3)2) + 2�4

2��t� (3� !)2 + 2�2

�2 ;

@p�A(!)

@!= t

2�4(3� 2!) + 6 (! � 3)2 t2�2 + ��2t(�3!2 + 26! � 39) + 2�A�(2�2(2� !)� (! � 3)2 t�)��t� (3� !)2 + 2�2

�2 ;

@p�B(!)

@!= �t(�3 (3� !)

2 t2�2 + t��2(!2 � 4! � 3) + 2�4 +�A�((3� !)2 t�+ 2�2))��t� (3� !)2 + 2�2

�2 .

Let us consider several cases:

� When �A = 0 then

@p�A(!)

@!= t

�2t�� �2

�(3t� (! � 3)2 � 2�2 (3� 2!))

(9t�� 2�2 � 6t�! + t�!2)2;

@p�B(!)

@!=3t2�2 (! � 3)2 � t��2

�!2 � 4! � 3

�� 2�4

(9t�� 2�2 � 6t�! + t�!2)2,

which means that @p�A(!)@! > 0 and @p�B(!)

@! < 0.

For R&D investments, we observe that

@x�A(!)

@!=�

�

� (9� !) (3� !)3 t2�2 + (7� 2!) (3� !)2 �t�2 � 2�4

(! � 3)2��t� (3� !)2 + 2�2

�2 ;

@x�B(!)

@!=�

�

6t2�2 (3� !)3 � (3� 2!) (3� !)2 t��2 � 2�4

(3� !)2��t� (3� !)2 + 2�2

�2 ;

which means that under most circumstances @x�A(!)@! < 0 and @x�B(!)

@! > 0.

@s�A(!)

@!=�3 (! � 3)2 t2�2 + t��2(!2 � 4! � 3) + 2�4

2��t� (3� !)2 + 2�2

�2 ;

which implies that when � is high in relation to t� then @s�A(!)@! > 0. Otherwise, @s

�A(!)@! < 0.

When �A 6= 0 then

20

sign

�@p�A(!)

@!

�= sign(2�4(3�2!)+6 (! � 3)2 t2�2+��2t(�3!2+26!�39)�2�A�((! � 3)2 t��2�2(2�!)));

sign(@p�B(!)

@!) = sign((3 (3� !)2 t2�2 � t��2(!2 � 4! � 3)� 2�4 ��A�((3� !)2 t�+ 2�2))),

sign(@s�A(!)

@!) = sign(�3 (! � 3)2 t2�2 + t��2(!2 � 4! � 3) + 2�4 +�A�(2�2 + t� (! � 3)2)):

Notice that sign(@p�B(!)@! ) = �sign(@s

�A(!)@! ), and

sign(@x�A(!)

@!) = sign(�t2�2 (9� !) (3� !)3 + (7� 2!) (3� !)2 �t�2 � 2�4);

@x�B(!)

@!= sign(6t2�2 (3� !)3 � (3� 2!) (3� !)2 t��2 � 2�4):

�

Proof of Corollary 2 The comparison of prices when the second order conditions are

satis�ed and �i = 0 is as follows:

p�i (0; 0)� p�B(!; 0) = �t!(3� !)t�+ �2

9t�� 2�2 � 6t�! + t�!2 < 0;

p�i (!;!)� p�i (0; 0) = t!

1� 2! > 0;

p�i (!;!)� p�B(!; 0) = t!t� (! + 2) (3� !)� �2 (3� 2!)

(1� 2!) (9t�� 2�2 � 6t�! + t�!2) ;

p�i (!;!)� p�A(!; 0) = t!t� (3! + 1) (3� !) + �2

�2!2 � 7! + 1

�(1� 2!) (9t�� 2�2 � 6t�! + t�!2) ;

p�A(!; 0)� p�i (0; 0) = t! (3� !)2t�� �2

9t�� 2�2 � 6t�! + t�!2 ;

p�A(!; 0)� p�B(!; 0) = t!t� (3� !)� �2 (4� !)9t�� 2�2 � 6t�! + t�!2 :

and the results from the Corollary follow since we also apply the second order conditions.

Let us now analyze the investment in R&D

21

sign(x�i (!;!)� x�i (0; 0)) = sign

0@�i ��9t�� 2�2

� �t� (2! � 3)2 � 2�2

�12t�2 (3� 2!) (3� !)

1A ;sign(x�A(!; 0)� x�B(!; 0)) = sign

��A �

t! (5� !)2

�;

sign(x�A(!; 0)� x�A(0; 0)) = sign �A �

�(3� !) (9� 2!) t�� �2

� �9t�� 2�2

�3t�2 (3� !) (6� !)

!;

and ((3�!)(9�2!)t���2)(9t��2�2)

3t�2(3�!)(6�!) > 0 is positive. Furthermore,

sign(x�B(!; 0)� x�B(0; 0)) = sign �9t�� 2�2

� �t� (3� !) (6� !) + �2

�3t�2 (! � 3) (! � 6)

��A

!;

sign(x�A(!; 0)�x�A(!;!)) = �sign

0@�(3� !) �!2 � 6! + 6� t�+ �2�+ 3t�2�A (2� !) (3� !) (3� 2!)�t� (2! � 3)2 � 2�2

�1A :

Then, we note that:

sign(x�B(!; 0)� x�B(!;!)) = sign

0@ �A+

+(2�4+t2�2(3�!)(!2�7!+9)(2!�3)2+t��2(2!3�24!2+72!�63))

3t�2(2�!)(3�!)(3�2!)

1A :Notice that when �A = 0 then x�B(!; 0)� x�B(!;!) = !�

(t�(3�!)(!2�7!+9)��2)�(3�2!)(3�!)(t�(2!�3)2�2�2)

> 0.

For the combined R&D of both �rms, we notice that for all �i: X�(!;!) �X�(0; 0) = �2!3�

�3�2! < 0, X

�(!; 0) �X�(!;!) = !��(3�2!)(3�!) > 0, and X

�(!; 0) �X�(0; 0) = �!�3�(3�!) < 0.

�

Proof of proposition 3 Let us �rst assume that �i = 0 then xA0 = xB0 and

TS�(0; 0) = xB0 +

�2�2

9�� t

4

�;

TS�(!;!) = xB0 +�2(1� !)� (3� 2!)

�2� !3� 2!

�� t

4;

22

TS�(!; 0) = xB0 +

+�t3�3

�2!2 � 6! + 9

�(3� !)4 + t��4

��2!3 + 5!2 + 24! � 36

�(! � 3)2

4� (3� !)2 (t� (3� !)2 � 2�2)2+

+4�6 (2� !) (4� !) + 2t2�2�2�!4 � 8!3 + 28!2 � 63! + 54

�(! � 3)2

4� (3� !)2 (t� (3� !)2 � 2�2)2:

Now, let us compute the di¤erences when �i = 0. We �rst note that

TS�(!;!)� TS�(0; 0) = �!�2(3� !)9� (2! � 3)2

< 0;

in fact note that @TS�(!;!)@! = ��2

�(3�2!)3 < 0. Hence,

TS�(!; 0)� TS�(0; 0) = !�4�6 (6� !) +�t��4

�18!2 � 77! � 24

�(! � 3)2

36� (! � 3)2 (9t�� 2�2 � 6t�! + t�!2)2+

+ !2t2�2�2

�18! � 24!2 + 5!3 � 27

�(! � 3)2 � 9t3�3! (! � 3)4

36� (! � 3)2 (9t�� 2�2 � 6t�! + t�!2)2:

Hence,

sign(TS�(!; 0)� TS�(0; 0)) =

sign(�4�6 (6� !)� t��4�18!2 � 77! � 24

�(! � 3)2

+2t2�2�2�5!3 � 24!2 + 18! � 27

�(! � 3)2 � 9t3�3! (! � 3)4):

The �rst, third and fourth terms are negative for the range of ! considered. The second

term is positive for the range of ! considered. Hence when � is large in relation to t� then

total surplus with asymmetric PPO may be larger than with no PPO, and hence also than with

symmetric PPO.

Another comparison is as follows:

TS�(!; 0)� TS�(!;!) =

!12�6 (2� !)� t��4

��93! + 126!2 � 60!3 + 8!4 + 24

�(! � 3)2

4� (2! � 3)2 (! � 3)2 (9t�� 2�2 � 6t�! + t�!2)2+

+!2t2�2�2

��36! + 25!3 � 14!4 + 2!5 + 27

�(! � 3)2 � t3�3! (2! � 3)2 (! � 3)4

4� (2! � 3)2 (! � 3)2 (9t�� 2�2 � 6t�! + t�!2)2:

Note that the �rst, second and third terms are positive while the fourth term is negative.

Then, if � is small in relation to t� are large then the last term could potentially dominate the

previous two terms. Hence, if some conditions are satis�ed then total surplus with asymmetric

PPO may be smaller than with symmetric PPO. �

23

Proof of Corollary 2 Let us �rst assume that �i 6= 0 then

TS�(0; 0) =1

2

�(xA0 + x

B0 ) +

3��2A9t�� 2�2

�+

�1 +

9�2�2A(9t�� 2�2)2

��2

9

�2

�� t

4

�;

TS�(!;!) =(xA0 + x

B0 )

2+�2(1� !)� (3� 2!)

�2� !3� 2!

�� t

4+

�2�2A (2! � 3)2

(t� (2! � 3)2 � 2�2)2

�2�2(1� !)� (2! � 3)2

� t

4

�+

��2A(3� 2!)2(t� (2! � 3)2 � 2�2)

:

Total surplus with asymmetric PPO is beyond the scope of this paper. The comparison of

the previous two suggests that

TS�(!;!)� TS�(0; 0) = � !�2(3� !)9� (2! � 3)2

+

+!�2i t�

2�4t��2

�4!3 + 12!2 � 54! + 27

��(9t�� 2�2)2 (t� (2! � 3)2 � 2�2)2

+

+!�2i t�

2�9t2�2 (6� 5!) (2! � 3)2 � 4�4

�4!2 � 13! + 12

��(9t�� 2�2)2 (t� (2! � 3)2 � 2�2)2

;

it sign can be re-written as

sign(TS�(!;!)� TS�(0; 0)) = sign��2i�� �

�where � =

�4t��2

�4!3 + 12!2 � 54! + 27

�+ 9t2�2 (6� 5!) (2! � 3)2 � 4�4

�4!2 � 13! + 12

��and � =

�2(3�!)(9t��2�2)2(t�(2!�3)2�2�2)2

9t�(2!�3)2 . �

Proof of Proposition 4. i) Symmetric PPO

We �rst need to calculate the second-best welfare optimal R&D investment taking as given

the equilibrium prices at t=2. As such, we have that s = �t(2!�3)+�A+�(xA�xB)2t(3�2!) , and hence

@s(!)@xA

= ��2t(2!�3) and

@s(!)@xB

= �2t(2!�3) :The �rst order condition is then

@TS

@xA=

@s

@xA�A + �(xA � xB)

@s

@xA+ �s� t

2(2s� 2(1� s)) @s

@xA� �xA;

and the second order condition is

@2TS

@x2A=�2(5� 4!)� 2t� (2! � 3)2

2t (2! � 3)2< 0;

which is equivalent to �2 < 2�t(2!�3)2(5�4!) and note that @2TS

@xA@xB= � �2(5�4!)

2t(2!�3)2 < 0. If the stability

condition �t (2! � 3)2 � �2(5 � 4!) > 0. Hence, we obtain the reaction functions in the �rst

period

24

xi =��i (5� 4!) + �t (2! � 3)2

2t� (2! � 3)2 � �2(5� 4!)� �2 (5� 4!)2t� (2! � 3)2 � �2(5� 4!)

xj :

Then, �nding the �xed point we obtain the second-best e¢ cient R&D investments:

x���i (!;!) =�

2�

�1 +

��i(5� 4!)(t� (2! � 3)2 � �2(5� 4!))

�As a special case, we �nd that with no PPO

x���i (0; 0) =�

2�

�1 +

��A(9t�� �2)

�

ii) Comparison of the equilibrium and e¢ cient R&D investments.

Comparing the expression of the second best optimal R&D and expression with the equilib-

rium one, we obtain that x�(!;!)�x���i (!;!) = ��t2�2(2!�3)4�2�4(5�4!)+t��2(7�4!)(2!�3)2�t�2�i(3�4!)(3�2!)3

2�(3�2!)(t�(2!�3)2�2�2)(t�(2!�3)2��2(5�4!)),

which implies

sign(x�i (!;!)� x���i (!;!)) = �sign(�si +�i)

where �si =(t�(2!�3)2�2�2)(t�(2!�3)2��2(5�4!))

t�2(3�4!)(3�2!)3 > 0.

With no-cross ownership we obtain that x�i (0; 0) � x���i (0; 0) = � �10�4+63t��2�81t�2(�i+t)

2�(3�2!)(9t��5�2)(9t��2�2) ,

which implies

sign(x�i (0; 0)� x���i (0; 0)) = �sign(�s0i +�i);

where �s0i =(9t��5�2)(9t��2�2)

81t�2> 0.

We also notice that do comparative statics of the second-best optimal quantity with respect

to !

sign(@x���i (!;!)

d!) = sign(�i):

This implies that when �i > 0 then increasing PPO increases the amount of e¢ cient R&D

investment.

iii) Asymmetric PPO. We �rst �nd the e¢ cient R&D investment under asymmetric PPO.

Note that

s =1

2t (3� !)(t(3� 2!) + �A + �(xA � xB))

and hence @s(!)@xA

= �2t(3�!) and

@s(!)@xB

= ��2t(3�!) . The �rst order condition is then

@TS

@xA=

@s

@xA�A + �(xA � xB)

@s

@xA+ �s� t

2(2s� 2(1� s)) @s

@xA� �xA

25

and hence

@TS

@xA=��A(5� 2!) + �2(xA � xB)(5� 2!) + t�

�2!2 � 8! + 9

�� 2t�xA (3� !)2

2t (3� !)2

the second order condition is satis�ed if

@2TS

@x2A=�2(5� 2!)� 2t� (3� !)2

2t (3� !)2< 0

and the stability condition requires that t�(3� !)2 � �2(5� 2!) > 0. Note that @2TS@xA@xB

=

��2(5�2!)2t(3�!)2 < 0. Hence, the best reaction function is

xA =��A(5� 2!) + t�

�2!2 � 8! + 9

�(2t� (3� !)2 � �2(5� 2!))

� �2(5� 2!)(2t� (3� !)2 � �2(5� 2!))

xB

For the acquired �rm we obtain the following

@TS

@xB=���A(5� 2!)2t (3� !)2

� �2(xA � xB)(5� 2!)

2t (3� !)2+�t(9� 4!)2t (3� !)2

� 2t� (3� !)2 xB

2t (3� !)2

@2TS

@x2B=�2(5� 2!)� 2t� (3� !)2

2t (3� !)2< 0

Then, from the �rst order condition, we obtain the reaction function

xB =���A(5� 2!) + �t(9� 4!)2t� (3� !)2 � �2(5� 2!)

� �2(5� 2!)2t� (3� !)2 � �2(5� 2!)

xA

Then, substituting one into the other, we obtain the e¢ cient R&D investments with asym-

metric PPO.

x���A (!; 0) = �(5� 2!)(��A � �2) + t�

�2!2 � 8! + 9

�2�(t� (3� !)2 � �2(5� 2!))

x���B (!; 0) = ��(��A + �2)(5� 2!) + t�(9� 4!)2� (t�(3� !)2 � �2(5� 2!)) :

The comparison leads to

sign(x�A(!; 0)� x���A (!; 0)) = sign(�aA ��A)

where �aA =�t2�2(2!+3)(3�!)3+t��2(!+1)(7�2!)(3�!)2�2�4(5�2!)

t�2(3�2!)(3�!)3 and

sign(x�B(!; 0)� x���B (!; 0)) = �sign (�B +�aB)

26

where �aB =t2�2(3�4!)(3�!)3+2�4(5�2!)�t��2(1�!)(7�2!)(3�!)2

t�2(3�2!)(3�!)3 :

Let us study the derivative of these two expressions with respect to !:

@x���A (!; 0)

d!=t���2(!2 � 5! + 5) + ��A (2� !) (3� !)� t� (3� 2!) (3� !)

�(t� (3� !)2 � �2(5� 2!))2

and

@x���B (!; 0)

d!=�t�

��2�!2 � 5! + 5

�+ ��A (2� !) (3� !)� t� (3� 2!) (3� !)

�(t� (3� !)2 � �2(5� 2!))2

Hence,

sign

�@x���A (!; 0)

d!

�= �sign

�@x���B (!; 0)

d!

�=

= sign��2(!2 � 5! + 5)� t� (3� 2!) (3� !) + ��A (2� !) (3� !)

��

27

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