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Product R&D Coopetition and Firm Performance

  • Michał Ramsza und Adam Karbowski ORCID logo EMAIL logo
Veröffentlicht/Copyright: 1. Februar 2020
The B.E. Journal of Theoretical Economics
Aus der Zeitschrift The B.E. Journal of Theoretical Economics Band 20 Heft 2

Abstract

We investigate firms’ behavior in demand-enhancing product R&D. We consider and compare a cooperative and non-cooperative R&D investment setting by firms. In a non-cooperative scenario (R&D competition), firms decide on their R&D investments and outputs unilaterally. In a cooperative scenario (R&D coopetition), firms engage in a bargaining process to reach a binding R&D agreement. Firms through bargaining can reach an R&D agreement which specifies their R&D investment levels. The investment levels under R&D coopetition are higher compared with the investment levels under R&D competition. Firms’ profits are also higher under R&D coopetition compared with R&D competition. We conclude that R&D coopetition can alleviate the individual R&D investment disincentive and work as a strategic instrument that enhances product innovation and firms’ profits.

Keywords: Product R&D; R&D investments; bargaining; firm performance

Acknowledgements

This research was supported by the National Science Centre, Poland, grant number 2016/21/B/HS4/03016 (Funder Id: http://dx.doi.org/10.13039/501100004281).

Appendix

A Proofs

The appendix contains proofs of all propositions.

Proof of Proposition 1. First order conditions read

∂π1∂q1=a+M(x1+x2)−c−2q1−q2=0,∂π2∂q2=a+M(x1+x2)−c−q1−2q2=0.

The above system of linear equations has a unique solution given by (3), that is a maximum since the payoffs (2) are concave. Given Assumption 1, Assumption 3 and Assumption 4, the values qi⋆>0, i = 1, 2.   □

Proof of Proposition 2. Firms’ payoffs at the market equilibrium qi⋆, i = 1, 2 read

πi(q1⋆,q2⋆,x1,x2)=19((a+M(x1+x2)−c)2−9xi),

for i = 1,2.

Nash equilibrium is derived through individual optimization of payoffs. First order conditions read

∂π1(q1⋆,q2⋆,x1,x2)∂q1=29M′(x1+x2)(a+M(x1+x2)−c)−1=0,∂π1(q1⋆,q2⋆,x1,x2)∂q1=29M′(x1+x2)(a+M(x1+x2)−c)−1=0.

The above equations are identical and so there is no possibility to derive individual investment levels. However, it is possible to find optimal total investment level γ⋆, that is, the positive solution of

(6)29M′(γ)(a+M(γ)−c)−1=0.

For γ⋆=0 eq. (6) becomes

∂π1(q1⋆,q2⋆,0,0)∂q1=29M′(0)(a−c)−1>0

due to the Assumption 3. On the other hand,

limγ→+∞29M′(γ)(a+M(γ)−c)−1<0

due to the Assumption 4. Also, due to the Assumption 2, the function 2M′(γ)(a+M(γ)−c)/9−1 is continuous and monotonically decreasing. Thus, there exists a unique, positive solution γ⋆ of eq. (6). The first part of Proposition 2 is proved.

To prove the second part, it is necessary to consider a set V of all viable outcomes of the bargaining problem. This set is defined as

V={(π1(q1⋆,q2⋆,x1,x2),π2(q1⋆,q2⋆,x1,x2))∈R2:x1≥0,x2≥0}∩R+2.

The set V is a family of curves indexed by x2 with the running parameter x1. Its Pareto boundary is (in part) an envelope line given as

π1(q1⋆,q2⋆,x1,x2)x1π2(q1⋆,q2⋆,x1,x2)x2−π1(q1⋆,q2⋆,x1,x2)x2π2(q1⋆,q2⋆,x1,x2)x1=0.

Simple algebra leads to the following equation

(7)(a+M(x1+x2)−c)M′(x1+x2)=94.

Equation (7) is similar to the eq. (6) that after rearranging reads

(8)(a+M(x1+x2)−c)M′(x1+x2)=92.

For the same reasons as before, there exists a unique, positive solution γc of (7). The proof is complete.   □

Proof of Proposition 3. The optimal total levels of R&D investment γ⋆ and γc are solutions of eq. (8) and (7), respectively. Since the left hand of (8), as well as (7), is a monotonically decreasing function, it is obvious that γ⋆<γc.

To show the second part of the proposition, the following inequality must be satisfied

π1(q1⋆,q2⋆,x1c,x2c)>π1(q1⋆,q2⋆,x1⋆,x2⋆).

The above inequality after substituting formulas in becomes

19((a+M(γc)−c)2−92γc)>19((a+M(γ⋆)−c)2−92γ⋆).

Simple algebra leads to

2(a−c)(M(γc)−M(γ⋆))+M(γc)2−M(γ⋆)2>0

but this inequality, due to the assumed conditions on M, a, c and the fact that γ⋆<γc, is clearly satisfied. The proof is complete.   □

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Published Online: 2020-02-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Research Articles
  2. The Effects of Entry when Monopolistic Competition and Oligopoly Coexist
  3. Managerial Delegation of Competing Vertical Chains with Vertical Externality
  4. Fiat Money as a Public Signal, Medium of Exchange, and Punishment
  5. Education Spending, Fertility Shocks and Generational Consumption Risk
  6. Should the Talk be Cheap in Contribution Games?
  7. College Assignment Problems Under Constrained Choice, Private Preferences, and Risk Aversion
  8. Competition with Nonexclusive Contracts: Tackling the Hold-Up Problem
  9. Endogenous Authority and Enforcement in Public Goods Games
  10. Disequilibrium Trade in a Large Market for an Indivisible Good
  11. Pretrial Beliefs and Verdict Accuracy: Costly Juror Effort and Free Riding
  12. Product R&D Coopetition and Firm Performance
  13. A Model of Inequality Aversion and Private Provision of Public Goods
  14. Managerial Accountability Under Yardstick Competition
  15. On the Equilibrium Uniqueness in Cournot Competition with Demand Uncertainty
https://doi.org/10.1515/bejte-2018-0141
Schlagwörter für diesen Artikel
Product R&D; R&D investments; bargaining; firm performance

Artikel in diesem Heft

  1. Research Articles
  2. The Effects of Entry when Monopolistic Competition and Oligopoly Coexist
  3. Managerial Delegation of Competing Vertical Chains with Vertical Externality
  4. Fiat Money as a Public Signal, Medium of Exchange, and Punishment
  5. Education Spending, Fertility Shocks and Generational Consumption Risk
  6. Should the Talk be Cheap in Contribution Games?
  7. College Assignment Problems Under Constrained Choice, Private Preferences, and Risk Aversion
  8. Competition with Nonexclusive Contracts: Tackling the Hold-Up Problem
  9. Endogenous Authority and Enforcement in Public Goods Games
  10. Disequilibrium Trade in a Large Market for an Indivisible Good
  11. Pretrial Beliefs and Verdict Accuracy: Costly Juror Effort and Free Riding
  12. Product R&D Coopetition and Firm Performance
  13. A Model of Inequality Aversion and Private Provision of Public Goods
  14. Managerial Accountability Under Yardstick Competition
  15. On the Equilibrium Uniqueness in Cournot Competition with Demand Uncertainty
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