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The Choice of Prices versus Quantities under Outsourcing

  • Ray-Yun Chang , Jin-Li Hu and Yan-Shu Lin EMAIL logo
Published/Copyright: January 13, 2018
The B.E. Journal of Theoretical Economics
From the journal The B.E. Journal of Theoretical Economics Volume 18 Issue 2

Abstract

This paper establishes a duopoly model with product differentiation and outsourcing in order to analyze the equilibrium competition strategies (choice of prices versus quantities) when the outsourcer outsources its intermediate good to a final product competitor. We show that: (1) both firms choose the quantity strategy when the cost efficiency of the subcontractor is low; (2) the choice of competition strategy is the price strategy for the subcontractor and the quantity strategy for the outsourcer when the cost efficiency of the subcontractor is moderate; (3) both firms choose the price strategy when the cost efficiency of the subcontractor is sufficiently high.

Keywords: competition strategy; outsourcing; intermediate market
JEL Classification: D43; L11; L13

Acknowledgements

We are grateful to two referees for their valuable comments, leading to substantial improvements of this paper. Financial support from the National Science Council (NSC-102-2410-H-259-003-MY3) is gratefully acknowledged. The usual disclaimer applies.

Appendix

A

According to the first-order conditions for profit maximization under the four possible strategy type combinations without outsourcing, we can obtain the equilibrium profits as follows:

π1QQ=[(a−c)(2−θ)+2δ4−θ2]2,
π2QQ=(q2QQ)2=[(a−c)(2−θ)−θδ4−θ2]2;
π1PQ=[(a−c)(2−θ−θ2)+δ(2−θ2)4−3θ2]2,
π2PQ=(1−θ2)(q2PQ)2=(1−θ2)[(a−c)(2−θ)−θδ4−3θ2]2;
π1QP=(1−θ2)[(a−c)(2−θ)+2δ4−3θ2]2,
π2QP=(q2QP)2=[(a−c)(2−θ−θ2)−θδ4−3θ2]2;
π1PP=1(1−θ2)[(a−c)(2−θ−θ2)+δ(2−θ2)4−θ2]2,
π2PP=(1−θ2)(q2PP)2=(1−θ2)[(a−c)(2−θ−θ2)−θδ(4−θ2)(1−θ2)]2.
B

According to the first-order conditions for profit maximization under the four possible strategy type combinations with outsourcing, we can obtain the equilibrium profits as follows:

π~1QQ=[2(a−c+δ)−θ(a−w)4−θ2]2+(w−c+δ)[2(a−w)−θ(a−c+δ)]4−θ2,
π~2QQ=[2(a−w)−θ(a−c+δ)4−θ2]2,
π~1PQ=[(2−θ2)(a−c+δ)−θ(a−w)4−3θ2]2+(w−c+δ)[2(a−w)−θ(a−c+δ)]4−3θ2,
π~2PQ=(1−θ2)[2(a−w)−θ(a−c+δ)4−3θ2]2,
π~1QP=[(2+2θ−2θ2)(a−c+δ)+θ3(a+c−2w−δ)−3θ(a−w)][(2−2θ)(a−c+δ)+θ(a−w)](4−3θ2)2+(w−c+δ)(1−θ)[2(a−w−θw)+θ(a+c−δ)]4−3θ2,
π~2QP=[(1−θ)[2(a−w−θw)+θ(a+c−δ)]4−3θ2]2,
π~1PP=[(2+2θ−θ2)(a−c+δ)−3θ(a−w)][2(a−c+δ)+θ(a−w)−θ2(w−c+δ)](1+θ)(4−θ2)2+(w−c+δ)[2(a−w−θw)+θ(a−c+δ)](1+θ)(4−θ2),
π~2PP=(1−θ)(1+θ)[2(a−w−θw)+θ(a−c+δ)4−θ2]2.
C

In this Appendix we allow the subcontractor to employ a two-part tariff (a fixed-fee F and a per-unit price w). Consequently, the profit functions of firms 1 and 2 under different combinations of strategy types are specified respectively as follows:

π~1kl=(p1−c+δ)q1+(w−c+δ)q2+F,
π~2kl=(p2−w)q2−F,k,l=P,Q.

We first examine the Q-Q game where the output levels are determined simultaneously. The third stage is the same as that in the Appendix B. We now move to the second stage of the game to derive the equilibrium fixed fee F and a per-unit price w. Firm 1’s profit function is expressed as:

maxw,F⁡ π~1QQ(q1QQ(w),q2QQ(w),w,F),
s. t. π~2QQ(w)−π2QQ(c)≥0, w>0, F≥0.

We assume that the subcontractor can extract from the outsourcer the entire rent accrued from outsourcing. Hence, the fixed fee charged by the subcontractor is defined as F(w)=(p2−w)q2QQ−π2QQ(c). After substituting F into π~1QQ, we differentiate π~1QQ with respect to w and set it equal to zero, the equilibrium per-unit price is thus yielded:

wQQ=θ(2−θ)2(a−c+δ)2(4−3θ2)+c−δ≡w^QQ.

After substituting w^QQ into F, the positive equilibrium fixed fee implies the range δ>[θ(2−θ)2(a−c)/(8−4θ−2θ2−θ3)]≡δQQ must be satisfied. If δ>δQQ, then a per-unit price w^QQ is an interior solution and a two part tariff is charged. When the first-order condition of profit maximization is larger than zero, firm 1 has an incentive to keep increasing the per-unit price. From the constraint F(w)=(p2−w)q2QQ−π2QQ(c)=0, we find that the per-unit price is w=c, which is a corner solution. Firm 1’s equilibrium fixed fee F and per-unit price w are expressed as:

(wQQ,FQQ)={wQQ = c≡w¯QQ, FQQ(w¯QQ)=0 ,δ≤δQQ;wQQ = θ(2−θ)2(a−c+δ)2(4−3θ2)+c−δ≡w^QQ, FQQ(w^QQ)>0 ,δ>δQQ.  

Similar to the above analysis, the equilibrium fixed fee and per-unit price under price-quantity competition are:

(wPQ,FPQ)={wPQ=c≡w¯PQ, FPQ(w¯PQ)=0 ,δ≤δPQ;wPQ=θ(a−c+δ)2+c−δ≡w^PQ, FPQ(w^PQ)>0 ,δ>δPQ, 

where

δPQ=θ(a−c)/(2−θ).

The equilibrium fixed fee and per-unit price under quantity-price competition are:

(wQP,FQP)={wQP=δ(2−θ2)2(1−θ2)+c−δ≡w¯QP , FQP(w¯QP)=0 δ≤δQP;wQP=θ(a−c+δ)2+c−δ≡w^QP, FQP(w^QP)>0 δ>δQP, 

where

δQP=θ(1−θ2)(a−c)/(2−θ−θ2+θ3).

Finally, the equilibrium fixed fee and per-unit price under price-quantity competition are:

(wPP,FPP)={wPP=δ(2−θ2)2(1−θ2)+c−δ≡w¯PP, FPP(w¯PP)=0 ,δ≤δPP;wPP=θ(2+θ)2(a−c+δ)2(4+5θ2)+c−δ≡w^PP, FPP(w^PP)>0 ,δ>δPP, 

where δPP=θ(2+θ)2(1−θ2)(a−c)/(8−4θ+2θ2+3θ3−θ4+θ5).

We now turn to the game’s first stage to determine the type of competition strategy. Since the subcontractor can extract from the outsourcer the entire rent accrued from outsourcing, the outsourcer’s profit is π2kl(c),k,l=P,Q. Given these Firm 2’s equilibrium profits, we obtain the relations that π2QQ(c)−π2QP(c)>0 and π2PQ(c)−π2PP(c)>0. Therefore, no matter what strategy type is adopted by firm 1, firm 2 has a dominant strategy to use the quantity strategy.

Given that quantity is the dominant strategy for firm 2, we have to determine the sign of π~1QQ(wQQ,FQQ)−π~1PQ(wPQ,FPQ). As δPQ>δQQ, we can discuss in the following three parts:

  1. When δ<δQQ, the equilibrium per-unit prices wQQ (=w¯QQ) and wPQ(=w¯PQ) are corner solutions, and the equilibrium fixed fee prices are FQQ(w¯QQ)=FPQ(w¯PQ)=0. Given these equilibrium intermediate good prices, we obtain the relations that π~1QQ(w¯QQ,FQQ)−π~1PQ(w¯PQ,FPQ)>0. Therefore, the equilibrium outcome is the quantity-quantity competition.

  2. When δQQ<δ<δPQ, the equilibrium per-unit price wQQ (=w^QQ) is an interior solution while that wQP(=w¯QP) is still a corner solution, and the equilibrium fixed fee prices are FQQ(w^QQ) and FPQ(w¯QP)=0. Given these equilibrium intermediate good prices, the sign of π~1QQ(w^QQ,FQQ)−π~1PQ(w¯PQ,FPQ) is indeterminate. Since π~1QQ(w^QQ,FQQ)−π~1PQ(w¯PQ,FPQ)=0, we can find that the critical value of δ is δTPT.[20] If δ<δTPT, then π~1QQ(w^QQ,FQQ)−π~1PQ(w¯PQ,FPQ)>0; otherwise, if δ>δTPT, then π~1QQ(w^QQ,FQQ)−π~1PQ(w¯PQ,FPQ)<0. Thus, when δ<δTPT(δ>δTPT), the equilibrium outcome is the quantity-quantity competition (price-quantity competition).

  3. When δ>δPQ, the equilibrium per-unit prices wQQ (=w^QQ) and wPQ (=w^PQ) are interior solutions, and the equilibrium fixed fee prices are FQQ(w^QQ) and FPQ(w^PQ). Given these equilibrium intermediate good prices, we obtain the relations that π~1QQ(w^QQ,FQQ)−π~1PQ(w^PQ,FPQ)<0. Therefore, the equilibrium outcome is the price-quantity competition.

In sum, when the subcontractor’s cost efficiency is low (δ<δTPT), both firms will choose the quantity strategy; when the subcontractor’s cost efficiency is high (δ>δTPT), the subcontractor will choose the price strategy while the outsourcer will choose the quantity strategy. Proposition 4 summarizes the above discussion.

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Published Online: 2018-01-13

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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  1. Special Section on Logic and the Foundations of Game and Decision Theory (LOFT12)
  2. Introduction to the Special Section on Logic and the Foundations of Game and Decision Theory (LOFT12)
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  5. Non-Congruent Views about Signal Precision in Collective Decisions
  6. Structural Control in Weighted Voting Games
  7. Agreeing to Disagree with Conditional Probability Systems
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  12. Endogenous Timing in Vertically-Related Markets
  13. Strategic Investment under Incomplete Information
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  15. Consumer Heterogeneity and Surplus under Two-Part Pricing
  16. Conflict and Competition over Multi-Issues
  17. Limited Liability and High Bids in English Auctions
  18. The Choice of Prices versus Quantities under Outsourcing
  19. Wars of Attrition with Endogenously Determined Budget Constraints
  20. Notes
  21. A Note on Cohabitation and Marriage
https://doi.org/10.1515/bejte-2016-0195
Keywords for this article
competition strategy; outsourcing; intermediate market

Articles in the same Issue

  1. Special Section on Logic and the Foundations of Game and Decision Theory (LOFT12)
  2. Introduction to the Special Section on Logic and the Foundations of Game and Decision Theory (LOFT12)
  3. Estimating the Use of Higher-Order Theory of Mind Using Computational Agents
  4. Beyond Coincidence: The Reasoning Process Underlying Utility Proportional Beliefs Process
  5. Non-Congruent Views about Signal Precision in Collective Decisions
  6. Structural Control in Weighted Voting Games
  7. Agreeing to Disagree with Conditional Probability Systems
  8. Research Articles
  9. Traps and incentives
  10. 10.1515/bejte-2016-0058
  11. Short Sale Constraints, Correlation and Market Efficiency
  12. Endogenous Timing in Vertically-Related Markets
  13. Strategic Investment under Incomplete Information
  14. Strategic Effects between Price-takers and Non-price-takers
  15. Consumer Heterogeneity and Surplus under Two-Part Pricing
  16. Conflict and Competition over Multi-Issues
  17. Limited Liability and High Bids in English Auctions
  18. The Choice of Prices versus Quantities under Outsourcing
  19. Wars of Attrition with Endogenously Determined Budget Constraints
  20. Notes
  21. A Note on Cohabitation and Marriage
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