Are Partial Ownership Arrangements Advantageous?
Wilson A. Alley
Faculty of Economics, Tohoku Gakuin UniversityAbstract
Using a two-stage differentiated-production Cournot duopoly model with technologicalspillovers, this paper investigates the effect partial ownership arrangements (POAs) among
rivals has on the market outcome when firms also compete in a non-production activity, R&D.
When products are differentiated, POAs do not affect the market outcome. As goods become
more homogenous, the effect POAs has is dependent on the size of technological spillovers.
When they are not large, POAs increase firms' R&D effort and output; when they are large,
POAs decrease output. POAs are also found to decrease firms' profit levels.
Keywords: partial ownership arrangements, R&D, technological spillovers, production
differentiation, Cournot competition.
JEL classification numbers: D43, L13, L15.

1. Introduction
Firms may acquire a proportion of their rivals' stock for a variety of reasons and this isbecoming increasingly common in a number of industries. Given that the purchase of a rival's
stock is not subject to the antitrust scrutiny that horizontal mergers are, the effectcross-shareholding among rivals has on the market outcome needs examination.
For many firms, partial ownership arrangements (POAs) allow them to gain many of thebenefits of a merger while still retaining their autonomy. They facilitate the transfer oftechnology and managerial skills between firms while allowing firms the opportunity tocapture some of the profits yielded by the transfer. When technological spillovers occur, firms
also have an incentive to enter into a POA to capture some of its rivals' profits yielded by its
own R&D activities. Accordingly, POAs may not only facilitate the transfer of technology,
but the existence of technological spillovers may also encourage them. Firms also have an
incentive to enter into a POA to reduce the amount of competition in the market to increase
profits. As the more homogenous the market, the more competitive the market is likely to be;
we would expect POAs are more likely to be formed to reduce market competition and in
more homogenous markets.
Given that firms compete not only in the product markets, but also in other non-production
activities such as research and development (R&D); inter-firm relationships that influencefirms' behavior in the product markets, is also likely to affect non-production activities. Thus
it is necessary to investigate not only how POAs affect production behaviour, but also theeffect it has on firms' non-production activities.
2. The basic model
We assume an industry that consists of two identical firms producing a differentiatedproduct that engage in upstream R&D and downstream production activities. It is assumed
each firm may also own a share of its rival. When it does, it is assumed to act as a silent(passive) partner. Each firm, however, also realizes that their output decisions affect theirrival and take this into consideration when choosing their output level. Firm i's output andR&D effort are denoted by qi and xi (i = 1,2) respectively. For simplicity cross-shareholdings
are assumed to be mutual and denoted by s; where 0 ≤ s ≤ 0.5 .
The representative consumer's utility function is assumed to be given by U = u (q1 , q2 ) + Z

1
where Z is a numeraire good. Assuming u (q1 , q2 ) = a(q1 + q2 ) − b(q12 + q22 + 2γ q1q2 ) , for (a,2b > 0) and the inverse of γ ∈ [0,1) indicates the degree of product differentiation. When γ = 1,firms produce homogeneous products; when γ = 0 products are independent. It is straightforward to derive the inverse demand functions:pi = a − b(qi + γ q j )(1)

for i,j = 1,2 and j≠i. Each Firm i's production cost function is assumed to take the form(c − xi + β x j )qi(2)where c is a positive constant and the parameter β ∈ [0,1] measures the spillover effect.
Accordingly, each firms R&D effort determines not only each firm's marginal cost, but also
the marginal cost of its rivals. When β = 0, Firm i's R&D affects only its own production cost
and the greater its research efforts, the lower its costs. When β = 1, spillovers are perfect andaccordingly, greater research efforts by Firm j lowers the costs of Firm i. The cost of Firm i'sR&D is assumed to be δ qi2 / 2 , where δ > 0 is a constant.

3. The two-stage game

As is standard, the model is solved backwards beginning from the second stage in whichfirms chose output. Assuming firms act in a Cournot fashion, then Firm i chooses its output
level qi to maximize profits (1-s)πi +sπi. It is straightforward to show that Firm i's output, as afunction of R&D efforts, is given byqi ( xi , x j ) =(1 − s ){2(1 − s ) − γ ) A + (2(1 − s ) − βγ xi + (2(1 − s ) β − γ ) x j }(2(1 − s ) − γ )(2(1 − s ) + γ )b(3)where A = a - c > 0.
Having derived the second-stage output as a function of R&D effort, we can now derive thesubgame perfect equilibria level of R&D effort, output and profits. This allows us to drop thesubscripts when denoting equilibrium values and assuming a symmetric equilibrium withregards to x, (3) simplifies toq ( x) =(1 − s ){ A + (1 + β ) x}.
(2(1 − s ) + γ )b(4)Using (3) we can now derive the subgame perfect equilibria values of per-firm R&D effort,
output and profits. These areAx = (1 − s )(2(1 − s) − γ ){2(1 − s ) 2 (2 − βγ ) + 2s (1 − s )γ − s (1 + β )γ 2 }
(5)

μq=

δA
(1 − s )(2(1 − s ) − γ )2 (2(1 − s ) + γ )μ(6)δ A2φπ= 2(7)μwhere μ = (2(1 − s ) − γ ) 2 (2(1 − s ) + γ ) 2 bδ − (1 − s)(1 + β ){4(2 − (1 + β )γ )(1 − s)3 − 2(1 − s )(2s − (1 − 2s ) β )γ 2 + s (1 + β )γ 3} and φ = (2(1 − s) − γ ) 2{(2(1 − s ) − γ ) 2 (2(1 − s) + γ ) 2 (1 − s )bδ−(1 − s ) 2 [2(1 − s ) 2 (2 − βγ ) + 2 s(1 − s)γ − s (1 + β )γ 2 ]2 / 2} .
How POAs, product differentiation and technological spillovers affect firms' R&D
activities, output levels and profits is shown graphically in Figure 1.
4. Conclusion

It was found when firms also compete in non-production activities such as R&D, regardlessof the level of POAs, when products are highly differentiated, larger technological spillovers
result in higher R&D, output and profit levels. As products become more homogenous, theeffect POAs have on R&D efforts and output is dependent on the size of technologicalspillovers. When they are not large, POAs have a positive effect; when they are large, POAs
have a negative effect on output levels. As POAs do not appear to increase firms' profit levels,from an antitrust perspective, POAs are advantageous in industries where products are nothighly differentiated and technological spillovers are not large as this results in greater R&D
efforts and higher output levels.
Main References

Brod, Andrew and Ram Shivakumar, 1999, `Advantageous Semi-collusion', Journal ofIndustrial Economics 47, 221-230.
Fershtman, Chaim and Neil Gandal, 1994, `Disadvantageous Semicollusion', International
Journal of Industrial Organization 12, 141-154.
Gilo, David, Yossi Moshe and Yossi Spiegel, 2006, `Partial Cross Ownership and tacit
Collusion', Rand Journal of Economics 37, 81-99.
Ono, Hiroshi, Takuya Nakazato, Colin Davis and Wilson Alley, 2004, `Partial Ownership
Arrangements in the Japanese Automobile Industry; 1990 - 2000', Journal of Applied
Economics 7, 355 -367.
Suzuki, Kazuyuki, 1993, `R&D Spillovers and Technology Transfer among and with Vertical
Keiretsu Groups. Evidence form the Japanese electrical machinery industry', International
Journal of Industrial Organization 11, 573-591.

Profits

OutputR&D
5435630525

2043γ = 0.1

153102251βs

01βs01.80s

β1.8

1.630

1.6251.4201.2151101.4

γ = 0.3

1.21βs

0.80.8βs1.20.61.25βs021.111.120.91

γ = 0.5

10.80.910.7
0.8

50.6βs0.7βs

βs

0110.910.880.74

1.81.6

γ = 0.7

1.41.21βs

β50.8s

064

1.5

121β

s

Figure 1: Relative levels of R&D, Output and Profits

0.5

10822ss2.53β

β30.64γ = 0.90.60βs0