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Efficient non-cooperative bargaining despite keeping strategic information private
Journal of Corporate Finance 42 (2017) 287–294
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Efficient non-cooperative bargaining despite keeping strategic information private Elmar Lukas ⁎, Andreas Welling Faculty of Economics and Management, Otto-von-Guericke University Magdeburg, Germany
a r t i c l e
i n f o
Article history: Received 1 February 2016 Received in revised form 13 December 2016 Accepted 14 December 2016 Available online 15 December 2016 JEL classification: G30 D81 L23 C70
a b s t r a c t The preferred method for negotiating joint value-added activities, such as e.g. licensing, co-venturing or supply chain expansion, is cooperative bargaining, where agents act rationally and have perfect information, i.e., information sharing occurs. In such settings, however, stronger partner often insist on keeping their valuable information private and thus cooperative bargaining cannot occur due to the lack of information sharing. Though, non-cooperative bargaining is usually assumed to be inefficient. Based on a game-theoretic real options model, we develop a contractual solution that allows the proposer to keep its valuable information private while at the same time motivates the other partner to act in a timely and efficient manner, that is, to be Pareto-efficient. © 2016 Elsevier B.V. All rights reserved.
Keywords: Optimal investment timing Real options Game theory Supply chain Negotiation
1. Introduction Traditionally, competition took place between individual companies, but there has been a fundamental change during the last few decades due to the increased use of outsourcing, licensing, merging, and collaboration. Today, competition has shifted from the individual-company level to encompass the entire network of economic value added (Lambert and Cooper, 2000). Apart from increased structural integration of upstream and downstream industry chains by means of mergers and acquisitions (M&A) the proper coordination of ongoing collaboration activities, for example among supply chain partners, is of central concern within the domain of operations research. Consequently, game theory should provide valuable insights in the decision-making regarding a common project of several companies. However, the bargaining companies do not only have to agree on how profits will be shared among them, but also about the timing of a common project. Recent literature acknowledges that classic net present value (NPV) is static in the sense that it requires agents to make decisions immediately (e.g., Keswani and Shackleton, 2006; Chevalier-Roignant et al., 2011). In contrast, viewing investment as an option right, that is, one has the right to invest but is not obliged to, makes it highly important to discover the optimal timing of an investment. As a result of prevailing uncertainties the investors face a tradeoff between early engagement, which ensures a possible first-mover advantage and allows generating profits early, and a later decision to maintain ⁎ Corresponding author at: Faculty of Economics and Management, Chair in Financial Management and Innovation Finance, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany. E-mail addresses: [email protected] (E. Lukas), [email protected] (A. Welling).
http://dx.doi.org/10.1016/j.jcorpfin.2016.12.004 0929-1199/© 2016 Elsevier B.V. All rights reserved.
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flexibility. This flexibility is important because of the irreversible nature of most strategic investments, i.e. financial commitments are at least partially sunk. They cannot fully be recovered if the project is abandoned at a later stage. Consequently, option-based valuation is a valuable tool to address these issues and has been successfully used in the literature to analyze the optimal timing of an investment under uncertainty (see e.g., Dixit and Pindyck, 1994; Trigeorgis, 1999; Kort et al., 2010; Hori and Osano, 2014). In particular, game theoretical decisions under uncertainty have been analyzed with the help of the real option framework in numerous applications, for example negotiating M&As (Lambrecht, 2004; Hackbarth and Miao, 2012; Lukas and Welling, 2012), formation and dissolution of a joint venture (Lee, 2004; Cvitanić et al., 2011; Martzoukos and Zacharias, 2013), collaboration in a supply chain (Amini and Li, 2011; Jörnsten et al., 2013; Lukas and Welling, 2014), as well as competition on IPO markets (Banerjee et al., 2016) or in an oligopoly (Bayer, 2007; Czarnitzki and Toole, 2013; Kamoto and Okawa, 2014). From a game-theoretic perspective two strands of literature have emerged on this topic. The first strand deals with the fact that the outcome of the negotiations about a common project is the result of a cooperative decision-making process. Here, the agents jointly maximize their profit in a cooperative game-theoretic manner. Collaboration can increase the effectiveness and thus the competitiveness of the collaborating companies (Lee, 2004; Pawlina, 2010; Cvitanić et al., 2011; Lukas and Welling, 2014). However, it requires common decision making, information and knowledge sharing, and the reduction (ideally, elimination) of opportunistic behavior (Maloni and Benton, 1997). Cooperative game-theoretic modeling requires that all agents involved in the negotiation have perfect information. However, companies often do not want to share their private information, maybe due to confidentiality concerns (Lee and Whang, 2000; van der Rhee et al., 2010). Many supply chains, for example, are dominated by a key company with superior negotiating power that allows to keep their own strategic information private though forcing their suppliers and retailers to disclose their private information (Lejeune and Yakova, 2005). The result is a vertical information asymmetry between the key company and the other companies in the supply chain (Li and Zhang, 2008). In contrast, the second strand of literature allows the agents to individually maximize their profits in the negotiation, leading to an application of non-cooperative game theory. However, it is well known that the outcome of non-cooperative managed projects is threatened by inefficiency, leading to inferior profit sharing, that is, Pareto-inefficient solutions. (see, e.g., Li, 2008). In particular, investment may be performed inefficiently late (Lambrecht, 2004; Cvitanić et al., 2011; Lukas and Welling, 2012, 2014) as well as inefficiently early (Bayer, 2007; Kamoto and Okawa, 2014). Building on these findings, the aim of this paper is to design a contract that combines aspects of cooperative and non-cooperative bargaining to overcome the specific problem that, on one hand, economically efficient investment can be achieved only by cooperative bargaining, while on the other hand non-temporal and non-cooperative bargaining assigns all the profit to the first mover, that is, the supply chain dominant firm. We show that such a contract design exists in a dynamic continuous-time setting and that the design has the added advantage of allowing the supply chain dominant firm to keep information private. As a result, our paper also contributes to the literature on principal-agent theory with one-sided incomplete information, i.e. when the principal has private information (see e.g. Myerson (199), Maskin and Tirole (1990, 1992)). In particular, our results fortify the findings of the traditional literature dealing with the informed-principal problem insofar as the principal, i.e. the dominant firm, does not perform worse than in the full-information case when retaining some discretion. The remainder of the paper is structured as follows. Section 2 sets out the general assumption and model structure. The analytical solutions for the extreme poles of negotiating a contract, that is, either completely non-cooperative or completely cooperative bargaining, are presented in Subsections 2.1 and 2.2 and serve as benchmarks for the later analysis. Subsection 2.3 defines the requirements for an optimal non-cooperative solution and Section 2.4 gives an example of such. Section 3 concludes and offers some suggestions for future research. 2. Preliminaries We consider a (potential) supply chain consisting of the multinational key company K and a local retailer R. The key-company produces a good with manufacturing costs of cK N 0 per unit and has established a binding retail price of pR N cK per unit. In t0 the key company has the opportunity to invest in the expansion of its business into a new foreign market. There, it depends on the cooperation of the local retailer R. The expansion requires the key company to make an investment of IK N 0. Likewise, the retailer has to make an investment of IR N 0.1 The (potential) revenues v(t) in the local market evolve stochastically over time due to overall economic uncertainty as well as due to uncertainties regarding customer's attitudes and behavior. In particular, we assume that they follow the geometricBrownian motion dvðt Þ ¼ μvðt Þdt þ σvðt ÞdW ðt Þ; vð0Þ ¼ v0 N0
ð1Þ
revenues, μR as its growth rate, and dW(t) with v0 N 0 as the (potential) revenues at time t0, σ≥ 0 as the volatility of the (potential) pffiffiffiffiffi as the increment of a Wiener process with zero mean and variance equal to dt which is correlated to the market portfolio. The degree of this correlation is assumed to be constant and measured by ρ ∈ (−1, 1). After the expansion in the local market, the keycompany's profit at time t ≥t0 equals pKp−cK vðtÞ. Here, pK ∈ [cK, pR] denotes the sales price to the retailer. Likewise, the retailers profit R at time t≥ t0 equals pR −ppK −cR vðtÞ with cR ∈ [0, pR − pK − cK] as the retailer's operational costs per unit sold. R
1 For example, the key-company has to bear costs for expanding its production and logistics network, while the retailer has to bear costs for preparing the distribution network.
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We assume that v0 , μ and σ derivate from market research done at t0 and hence are known by both companies.2 However, after t0 and before the expansion takes place only the retailer is able to estimate the current (potential) revenues in the local market. After the expansion, the exact revenues are directly observable by the retailer, while they are still unobservable for the keycompany, which at most may observe the orders of the retailer. Both companies are risk-averse and base their calculations on the CAPM formula (see Merton (1973)), with r ≥μ −λρσ as the riskless interest rate and λR as the market price of risk which is assumed to be constant. Thus, the expected present value of the vðtÞ key-company's future cash flow stream at time t ≥t0 equals pKp−cK rþλρσ −μ , which we denote as the key-company's present value. R vðtÞ pR −pK −cR : Likewise, the retailer's present value equals p rþλρσ−μ R
2.1. Efficient investment timing: the cooperative solution Assume that both companies are owned by the same external decision maker. Consequently, the aggregated profit he makes when investing at an arbitrary time, i.e., τ≥ t0 results in: −r ðτ−t 0 Þ
πE ðτÞ ¼ e
pR −cK −cR vðt Þ −I K −IR : pR r þ λρσ −μ
ð2Þ
Following Dixit and Pindyck (1994), the decision to invest can be regarded as a real option that should be exercised at the efficient investment time τE⁎ = inf{t≥ t0 | v(t) ≥vE⁎}, which is determined by the efficient investment threshold vE ¼
β ðr þ λρσ −μ ÞðIK þ IR ÞpR ; pR −cK −cR β−1
β¼
1 μ−λρσ þ − 2 σ2
ð3Þ
where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 1 μ−λρσ 2 2r þ 2: − 2 2 σ σ
ð4Þ
Following this rule, the expected profit of the external decision maker equals 8 β ðpR −cK −cR ÞvE v0 > > −ðIK þ IR Þ < p ð r þ λρσ −μ Þ vE R GE ¼ ð p −c −c Þv > R K R 0 > : −ðI K þ IR Þ pR ðr þ λρσ−μ Þ
vE Nv0
ð5Þ
vE ≤v0
2.2. Taking full advantage of being the first mover: the non-cooperative solution Now, we assume a sequential bargaining process that, unlike cooperative bargaining, allows the key company to keep information about cK and IK confidential.3 First, at time t0 the key company offers a sales price pK N 0 to the retailer. Then, the retailer immediately decides whether it will accept or reject the offer. In such a static ultimatum game setting the key company gains πU;K ðpK Þ ¼
ðpK −cK Þv0 −I pR ðr þ λρσ −μ Þ K
ð6Þ
with the expansion of its business, while the retailer gains πU;R ðpK Þ ¼
ðpR −pK −cR Þv0 −I : pR ðr þ λρσ−μ Þ R
ð7Þ
Hence, the key company could make an offer pK,U⁎ that is the highest the retailer could accept, i.e. πU,R(pK,U⁎) = 0, and that simultaneously would not represent a loss to the key company itself, i.e., πU, K(pK,U⁎) ≥ 0. Thus, the parties could possibly agree on the þI R ÞpR ðr þ λρσ −μÞ ¼: vU : Consequently, the optimal offered sales price would bepK;U ¼ pR −cR − terms of the deal if v0 ≥ ðpðIK−c K −cR Þ R I R pR ðrþλρσ−μÞ : However, it is well known that in such a setting the key company receives the entire surplus whereas the retailer v0 receives nothing. Thus, the retailer is indifferent between accepting the offer and rejecting it. According to Güth et al. (1982) the key-company should therefore at least leave the minimal positive amount to the retailer. Besides that, the key company 2 Any potential costs for doing this market research would be sunk costs as they have already have been borne by the key-company before t0. Consequently, they can be ignored in our model. 3 In particular, the retailer is not able to verify the exact values of cK and IK from the offered sales price pK,U⁎.
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may wish to grant a little profit of x N 0 to its retailer to keep the retailer loyal and economically viable. Consequently, the offered −μÞx and hence becomes pK;U ¼ pR −cR − ðIR −xÞpR ðrþλρσ−μÞ . For the total surplus in the ultimatum sales price decreases by pR ðrþλρσ v0 v0 game we obtain ðp −c −c Þv GU ¼ max 0; R R K 0 −I K −IR : pR ðr þ λρσ−μ Þ
ð8Þ
Hence, as long as vE⁎ N v0 we get that GU b GE. This overall loss is due to losing the value of being able to defer the investment. Consequently, in the ultimatum game, investment usually occurs inefficiently early.
2.3. Requirements on an optimal contractual solution As we have seen in the previous sub sections from the key-company's perspective cooperative bargaining as well as non-cooperative bargaining via the ultimatum game each comes with certain advantages and disadvantages. While the outcome of cooperative bargaining is Pareto-optimal, i.e. the total surplus is maximized due to efficient investment timing, the ultimatum game allows the key-company to keep the total surplus and to keep its information on cK and IK private. Consequently, the question arises if a contractual solution exists that combines the described advantages of the key-company after cooperative bargaining and non-cooperative bargaining. If vE⁎ ≤ v0 it is obvious that the ultimatum game already leads to the optimal outcome for the key company because it maximizes the total surplus (an external decision maker would also invest at t0) and grants it entirely to the key company. Furthermore, the key-company can keep its information on cK and IK private. Hence, in this case the keycompany does not need a special contract to improve its situation. In the following we, therefore, assume that vE⁎ N v0. In particular, an optimal contractual solution has to meet the following four requirements: (i) (ii) (iii) (iv)
The total surplus of the deal is maximized, i.e. investment timing is efficient. The key-company can decide alone on the division of the surplus, i.e. it can choose the expected profit x of its retailer. The key-company can keep its information on cK and IK private. The contractual solution does not presuppose that the revenues at the time of investment or at any later point in time are verifiable by the key-company or that both parties could make any contracts on their basis.
While the first requirement is self-explanatory the second requirement not only enables the key-company to keep the entire surplus but also allows him to voluntarily grant the retailer a part of this profit. Reasons for providing such a voluntarily grant might, e.g., be justified in order to perpetuate the loyalty of the retailer, to secure the economic viability of the retailer, or to improve the image of the company towards its current retailers as well as to possible future retailers. Advantageously, the contract proposed in this article does not provide an economic rule to decide on an optimal grant to the retailer. Rather it allows to exogenously optimizing for it as the resulting investment time in the proposed contract does not depend on the amount of this grant. The third requirement guarantees that the retailer does not get to know the exact values of cK and IK. This possibility to keep some information private may be advantageous to the key-company.4 For example, due to strategic considerations or because the key-company has committed to a third party not to share this information (Li and Zhang, 2008), or as without knowing the exact values of cK and IK the retailer is not able to determine the key-company's profit of the investment.5 The fourth requirement results from the situation that the realized revenues in the local market are not verifiable by the key-company and thus cannot be part of the contractual solution.
2.4. An optimal contractual solution In the following proposition we provide an optimal contractual solution, i.e. a solution that meets all the four requirements described above. Thereby, we build on the approaches of Lambrecht (2004) and Lukas and Welling (2012) who both describe a non-cooperative option game in which one party (the first-mover) offers the other party (the second-mover) a certain payment and then the second-mover has the right to decide on the investment-timing. However, if the first-mover simply offers an equity stake, i.e. percentage participation on the realized revenue, the second-mover will time the investment inefficiently (at least from the first-movers perspective). Likewise, offering a fixed payment does not lead to efficient investment timing, either. Surprisingly, an ideal combination of an “equity” stake and a fixed payment exists that leads to efficient investment timing and simultaneously meets the other two requirements described above:
4 Obviously, the contractual solution of this article does not prohibit the key-company to share all of its private information, but gives the key-company some flexibility regarding the information sharing, without jeopardizing the social-efficiency of the negotiation. 5 Consequently, the retailer cannot compare the key-company's profits with his own profits and therefore is not able to measure the fairness of the deal.
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2.4.1. Proposition: It is optimal to the key-company to let the retailer decide on the investment time τ ≥t0 and to offer him a sales price of
pK;C ðxÞ ¼ pR −cR −ðβ−1Þ
pR −cK −cR v0 −β x ðIR þ IK Þ vE
ð9Þ
in combination with a fixed payment at time τ of −β v x: ψK;C ðxÞ ¼ IR þ ð1−βÞ 0 vE
ð10Þ
2.4.2. Proof From the retailer's perspective the right to time the investment has to be regarded as a real option of perpetual American type. Exercising the option at time τ the retailer gains. 0 1 p −c −c −β ðβ−1Þ R K R pR − pK;C ðxÞ−cR ðIR þ IK Þ B C v vðτÞ þ ψK;C ðxÞ−IR ¼ @ vðτÞ þ ð1−βÞA 0 x πR ðτ; xÞ ¼ pR ðr þ λρσ−μ Þ pR ðr þ λρσ−μ Þ vE −β x v0 ¼ βvðτÞ−ðβ−1ÞvE vE vE
ð11Þ
Hence, the retailer will choose the investment time τC⁎ that maximizes its expected profit discounted on time t0. Thus, τC⁎ is given by −β −β v0 x v0 x −rðτC −t0 Þ −r ðτC −t 0 Þ −r ðτ−t 0 Þ ¼ max E e E βv τC −ðβ−1ÞvE e βvðτ Þ−ðβ−1ÞvE e vE τ∈½t 0 ;∞ vE vE vE
ð12Þ
From Dixit and Pindyck (1994) we know that it is optimal for the retailer to invest as soon as the value of the traded asset reaches a certain threshold vC⁎, i.e.
τC ¼ inf t ≥t 0 jvðt Þ≥vC ;
ð13Þ
and that the value of the retailer's option to invest can be calculated by f R vC ¼
−β β v0 x v0 : βvC −ðβ−1ÞvE vE vC vE
ð14Þ
Due to
∂f R ðvC Þ ¼ ∂vC
−β β β −β β v0 x v0 v0 v x v0 ððβ−1ÞÞ β − βvC −ðβ−1ÞvE ¼ − 0 vC −vE β β vE vC vE vC vE vC vC vE vC
ð15Þ
we can easily see that vC⁎ = vE⁎ is the unique solution of the equation
∂f R ðvC Þ ¼ 0: ∂vC
ð16Þ
Thus, the retailer will exercise the option at the efficient investment time. Considering that vC⁎ = vE⁎ we get from Eq. (14) that f R vC ¼
−β β v0 x v0 ¼ x: βvC −ðβ−1ÞvE vE vC vE
ð17Þ
Consequently, the key-company can solely decide to offer a certain profit with expected present value x N 0 to the retailer.6 Furthermore, the incentive contract has the advantage of keeping information private. By “private” information we mean that it is not possible for the retailer to discover the key company's strategic variables from the offered premium. As vE⁎ is known, the retailer can according to Eq. (3) determine a linear relationship between cK and IK. However, the size of vE⁎ cannot be used to gain full 6 In this regard, it should be noted that for x=0 the retailer would be indifferent between investing at any point in time and never investing, hence the key-company should - in analogy to the ultimatum game - at least offer the minimal positive amount x to the retailer.
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information regarding the key company's true investment cost IK nor its true size of manufacturing costs cK. In particular, for every z N 0 we get that |{(IK,cK)|vE⁎(IK,cK) = z}| = ∞. Finally, in the contractual solution (see Eqs. (9) and (10)) the key-company does not offer anything that would depend on the revenue at τ or at a later point in time □ In the ultimatum game the retailer has to decide on a take-it-or-leave-it base. In contrast, the contractual solution described above seemingly provides some flexibility to the retailer, i.e. he can wait with the decision and choose the individually optimal investment time. Though, the contract is designed in a way that (except of a voluntarily payment) only the key-company generates a surplus with the investment. Hence, it is the key-company and not the retailer who profits from the arising flexibility value. This result is achieved by offering a cost-plus contract (see for example Agrell et al., 2004, p. 6) where on one hand the retailer is investment-time-independently compensated with his investment costs IR and operating costs cR and on the other hand an additional voluntarily payment is designed in a way that its expected present value is maximized when the investment takes place at the socially-efficient investment time τE⁎. 2.5. The benefit of the optimal contractual solution In the following we analyze under which circumstances the contractual solution generates the highest benefit compared to the ultimatum game. Therefore, we first calculate this benefit, i.e. the difference of total gains from the contractual solution and the total gains from the ultimatum game: 8 β ðpR −cK −cR ÞvE v0 > > > −ðI K þ IR Þ > > p ð r þ λρσ−μ Þ v > > R E β > > v0 < ðpR −cK −cR ÞvE −ðI K þ IR Þ GE −GU ¼ pR ðrþ λρσ−μ Þ vE > > > ð Þv p −c −c > R R K 0 > −I −I − > > > pR ðr þ λρσ−μ Þ K R > : 0
v0 ≤vU
vU bv0 bvE
:
ð18Þ
vE ≤v0
In particular, we have to distinguish three cases. First, if the outcome of the contractual solution is not ultimate investment and the outcome of the ultimatum game would be no investment, i.e. for v0 ≤ vU⁎, the ultimatum gain generates no profit. Consequently, the benefit of the contractual solution compared to the ultimatum game equals the central planner's value of the option to invest. As can be easily seen from Eq. (18) this option value is increasing with increasing v0. Second, if the outcome of the contractual solution is not ultimate investment and the outcome of the ultimatum game would be investment, i.e. for vU⁎ b v0 b vE⁎, the benefit of the contractual solution compared to the ultimatum game is the difference of the central planner's value of the option to invest and the intrinsic value of this option. That is the extrinsic value of this option. However, as the total gain of the ultimatum game
Fig. 1. The additional value GE −GU of the contractual solution compared to the ultimatum game in dependence of the initial value v0 of the potential revenues and for different values of uncertainty and correlation to the market portfolio (fat line (benchmark case): σ=0.2,ρ=0; solid line (higher uncertainty): σ=0.3,ρ=0; dash-dotted line (lower uncertainty): σ= 0.1, ρ = 0; dashed line (negative correlation): σ = 0.2, ρ = − 0.2; dotted line (positive correlation): σ= 0.2, ρ = 0.2). Other parameter values: r=0.1,μ=0.05,λ=0.2,IK =10,IR =10, cK =1,cR =1,pR =4.
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increases faster in v0 than the central planner's option value (see Eq. (18)) the benefit of the contractual solution compared to the ultimatum game is decreasing in v0. Third, if the outcome of the contractual solution is ultimate investment, i.e. for vE⁎ ≤v0, its total gain equals the total gain of the ultimatum game which due to vE⁎ N vU⁎ leads to ultimate investment as well. Consequently, the benefit of the contractual solution compared to the ultimatum game is zero. Finally, we can therefore conclude that the benefit of the contractual solution compared to the ultimatum game is the highest, if the investment breaks even in the ultimatum game, i.e. if v0 =vU⁎ (see also Fig. 1). Fig. 1 furthermore depicts that the benefit GE − GU of the contractual solution in comparison to the ultimatum game increases with the uncertainty σ about the value of the traded asset. This is due to the flexibility value of the contractual solution that increases with uncertainty while the profit of the ultimatum game is not influenced by the uncertainty as long as the traded asset is uncorrelated to the market portfolio. However, a positive [negative] correlation of the traded asset and the market portfolio leads to a lower [higher] break-even point vU⁎ and a lower [higher] benefit of the contractual solution compared to the ultimatum game. Finally, another important result is that retaining information does not put the dominant firm in a worsen position compared to the full-information case. Rather, the outcome of an asymmetric Nash-bargaining game will assign less surplus assigned to the dominant firm than in the one-sided incomplete information case. To see this, we have to compare the full-information case, which is represented by Eq. (5) with the surplus generated by means of the optimal contract. Obviously, both strategies yield the same surplus with the only difference that the bargaining power determines how much surplus the dominant firm receives from the joint investment. Fig. 2 depicts the influence of several other parameters on the benefit of the contractual solution compared to the ultimatum game. In particular, a higher riskless interest rate leads to a higher break-even point vU⁎ and a lower benefit of the contractual solution compared to the ultimatum game. Contrarily, a higher growth rate of the revenues leads to a lower break-even point vU⁎ and a higher benefit of the contractual solution compared to the ultimatum game. A higher retail price also leads to a lower break-even point, but does hardly influence the benefit of the contractual solution. Finally, higher investment costs lead to a higher break-even point and come with a greater benefit of the contractual solution. 3. Conclusion Negotiating contracts is at the heart of every supply chain design. If structured cooperatively each party has to make its information public in order to jointly maximize profits and equitable profit sharing. If, however, the negotiations are structured noncooperatively profit sharing usually is assumed to be biased in favor of the first mover but inefficiencies arise due to postponement of strategic investment decisions. A natural question arises as to whether it is possible to design a contract that combines the positive aspects of both types of negotiating, i.e. cooperative and non-cooperative. In this paper, we show that such is indeed possible and when drawn up properly, such a contract provides an incentive to not further delay investment. Moreover, the
Fig. 2. The additional value GE −GU of the contractual solution compared to the ultimatum game in dependence of the initial value v0 of the traded asset and for different parameter values (fat line (benchmark case); solid line (higher retail price): pR =5; dash-dotted line (higher interest rate): r=0.12; dashed line (higher growth rate): μ=0.07; dotted line (higher investment costs): IK =15). Benchmark parameter values: σ=0.2,ρ=0,r=0.1,μ=0.05,λ=0.2,IK =10,IR =10, cK = 1,cR =1,pR =4.
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supply chain dominating firm can fully capitalize on the surplus generated - even if it is not able to observe variations in the generated revenues. Apart from the incentive mechanism and preservation of the first-mover advantage, the supply chain dominating firm can also keep private information on key strategic variables, e.g., the true size of its investment or size of realized synergies. One direction in which this work can be extended is to consider several rounds of renegotiation or to allow contract specifications to be changed in later rounds. Another idea is to include certain other sources of risk, for example, exchange rate or technological risks and investigate how these risks affect timing and optimal contract design. References Agrell, P.J., Lindroth, R., Norrman, A., 2004. Risk, information and incentives in telecom supply chains. Int. J. Prod. Econ. 90 (1), 1–16. Amini, M., Li, H., 2011. 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An experimental analysis of ultimatum bargaining. J. Econ. Behav. Organ. 3 (4), 367–388. Hackbarth, D., Miao, J., 2012. The dynamics of mergers and acquisitions in oligopolistic industries. J. Econ. Dyn. Control. 36 (4), 585–609. Hori, K., Osano, H., 2014. Investment timing decisions of managers under endogenous contracts. J. Corp. Finan. 29, 607–627. Jörnsten, K., Lise Nonås, S., Sandal, L., Ubøe, J., 2013. Mixed contracts for the newsvendor problem with real options and discrete demand. Omega 41, 809–819. Kamoto, S., Okawa, M., 2014. Market entry, capacity choice, and product differentiation in duopolistic competition under uncertainty. Manag. Decis. Econ. 35, 503–522. Keswani, A., Shackleton, M.B., 2006. How real option disinvestment flexibility augments project NPV. Eur. J. Oper. Res. 168 (1), 240–252. Kort, P.M., Murto, P., Pawlina, G., 2010. Uncertainty and stepwise investment. Eur. J. Oper. Res. 202 (1), 196–203. Lambert, D.M., Cooper, M.C., 2000. Issues in supply chain management. Ind. Mark. Manag. 29 (1), 65–83. Lambrecht, B.M., 2004. The timing and terms of mergers motivated by economies of scale. J. Financ. Econ. 72 (1), 41–62. Lee, T., 2004. Determinants of the foreign equity share of international joint ventures. J. Econ. Dyn. Control. 28 (11), 2261–2275. Lee, H.L., Whang, S., 2000. nformation sharing in a supply chain. Int. J. Technol. Manag. 20 (3/4), 373–387. Lejeune, M.A., Yakova, N., 2005. On characterizing the 4C′s in supply chain management. J. Oper. Manag. 23 (1), 81–100. Li, C.-L., 2008. Quantifying supply chain ineffectiveness under uncoordinated pricing decisions. Oper. Res. Lett. 36 (1), 83–88. Li, L., Zhang, H., 2008. Confidentiality and information sharing in supply chain coordination. Manag. Sci. 54 (8), 1467–1481. Lukas, E., Welling, A., 2012. Negotiating M&As under uncertainty: the influence of managerial flexibility on the first-mover advantage. Financ. Res. Lett. 9 (1), 29–35. Lukas, E., Welling, A., 2014. Timing and eco(nomic) efficiency of climate-friendly investments in supply chains. Eur. J. Oper. Res. 233 (2), 448–457. Maloni, M.J., Benton, W.C., 1997. Supply chain partnerships: opportunities for operations research. Eur. J. Oper. Res. 101 (3), 419–429. Martzoukos, S.H., Zacharias, E., 2013. Real option games with R&D and learning spillovers. Omega 41 (2), 236–249. Maskin, Eric, Tirole, Jean, 1990. “The principal-agent relationship with an informed principal. I: The Case of Private Values”. Econometrica 58 (2), 379–409. Maskin, Eric, Tirole, Jean, 1992. “The principal-agent relationship with an informed principal, II: Common Values”. Econometrica 60 (1), 1–42. Merton, R.C., 1973. An intertemporal capital asset pricing model. Econometrica 41 (5), 867–887. Pawlina, G., 2010. Underinvestment, capital structure and strategic debt restructuring. J. Corp. Finan. 16, 679–702. Trigeorgis, L., 1999. Real Options: Managerial Flexibility and Strategy in Resource Allocation. fourth ed. MIT Press, Cambridge. Van der Rhee, B., Van der Veen, J.A.A., Venugopal, V., Nalla, V.R., 2010. A new revenue sharing mechanism for coordinating multi-echelon supply chains. Oper. Res. Lett. 38 (4), 296–301.
Contents lists available at ScienceDirect
Journal of Corporate Finance journal homepage: www.elsevier.com/locate/jcorpfin
Efficient non-cooperative bargaining despite keeping strategic information private Elmar Lukas ⁎, Andreas Welling Faculty of Economics and Management, Otto-von-Guericke University Magdeburg, Germany
a r t i c l e
i n f o
Article history: Received 1 February 2016 Received in revised form 13 December 2016 Accepted 14 December 2016 Available online 15 December 2016 JEL classification: G30 D81 L23 C70
a b s t r a c t The preferred method for negotiating joint value-added activities, such as e.g. licensing, co-venturing or supply chain expansion, is cooperative bargaining, where agents act rationally and have perfect information, i.e., information sharing occurs. In such settings, however, stronger partner often insist on keeping their valuable information private and thus cooperative bargaining cannot occur due to the lack of information sharing. Though, non-cooperative bargaining is usually assumed to be inefficient. Based on a game-theoretic real options model, we develop a contractual solution that allows the proposer to keep its valuable information private while at the same time motivates the other partner to act in a timely and efficient manner, that is, to be Pareto-efficient. © 2016 Elsevier B.V. All rights reserved.
Keywords: Optimal investment timing Real options Game theory Supply chain Negotiation
1. Introduction Traditionally, competition took place between individual companies, but there has been a fundamental change during the last few decades due to the increased use of outsourcing, licensing, merging, and collaboration. Today, competition has shifted from the individual-company level to encompass the entire network of economic value added (Lambert and Cooper, 2000). Apart from increased structural integration of upstream and downstream industry chains by means of mergers and acquisitions (M&A) the proper coordination of ongoing collaboration activities, for example among supply chain partners, is of central concern within the domain of operations research. Consequently, game theory should provide valuable insights in the decision-making regarding a common project of several companies. However, the bargaining companies do not only have to agree on how profits will be shared among them, but also about the timing of a common project. Recent literature acknowledges that classic net present value (NPV) is static in the sense that it requires agents to make decisions immediately (e.g., Keswani and Shackleton, 2006; Chevalier-Roignant et al., 2011). In contrast, viewing investment as an option right, that is, one has the right to invest but is not obliged to, makes it highly important to discover the optimal timing of an investment. As a result of prevailing uncertainties the investors face a tradeoff between early engagement, which ensures a possible first-mover advantage and allows generating profits early, and a later decision to maintain ⁎ Corresponding author at: Faculty of Economics and Management, Chair in Financial Management and Innovation Finance, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany. E-mail addresses: [email protected] (E. Lukas), [email protected] (A. Welling).
http://dx.doi.org/10.1016/j.jcorpfin.2016.12.004 0929-1199/© 2016 Elsevier B.V. All rights reserved.
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flexibility. This flexibility is important because of the irreversible nature of most strategic investments, i.e. financial commitments are at least partially sunk. They cannot fully be recovered if the project is abandoned at a later stage. Consequently, option-based valuation is a valuable tool to address these issues and has been successfully used in the literature to analyze the optimal timing of an investment under uncertainty (see e.g., Dixit and Pindyck, 1994; Trigeorgis, 1999; Kort et al., 2010; Hori and Osano, 2014). In particular, game theoretical decisions under uncertainty have been analyzed with the help of the real option framework in numerous applications, for example negotiating M&As (Lambrecht, 2004; Hackbarth and Miao, 2012; Lukas and Welling, 2012), formation and dissolution of a joint venture (Lee, 2004; Cvitanić et al., 2011; Martzoukos and Zacharias, 2013), collaboration in a supply chain (Amini and Li, 2011; Jörnsten et al., 2013; Lukas and Welling, 2014), as well as competition on IPO markets (Banerjee et al., 2016) or in an oligopoly (Bayer, 2007; Czarnitzki and Toole, 2013; Kamoto and Okawa, 2014). From a game-theoretic perspective two strands of literature have emerged on this topic. The first strand deals with the fact that the outcome of the negotiations about a common project is the result of a cooperative decision-making process. Here, the agents jointly maximize their profit in a cooperative game-theoretic manner. Collaboration can increase the effectiveness and thus the competitiveness of the collaborating companies (Lee, 2004; Pawlina, 2010; Cvitanić et al., 2011; Lukas and Welling, 2014). However, it requires common decision making, information and knowledge sharing, and the reduction (ideally, elimination) of opportunistic behavior (Maloni and Benton, 1997). Cooperative game-theoretic modeling requires that all agents involved in the negotiation have perfect information. However, companies often do not want to share their private information, maybe due to confidentiality concerns (Lee and Whang, 2000; van der Rhee et al., 2010). Many supply chains, for example, are dominated by a key company with superior negotiating power that allows to keep their own strategic information private though forcing their suppliers and retailers to disclose their private information (Lejeune and Yakova, 2005). The result is a vertical information asymmetry between the key company and the other companies in the supply chain (Li and Zhang, 2008). In contrast, the second strand of literature allows the agents to individually maximize their profits in the negotiation, leading to an application of non-cooperative game theory. However, it is well known that the outcome of non-cooperative managed projects is threatened by inefficiency, leading to inferior profit sharing, that is, Pareto-inefficient solutions. (see, e.g., Li, 2008). In particular, investment may be performed inefficiently late (Lambrecht, 2004; Cvitanić et al., 2011; Lukas and Welling, 2012, 2014) as well as inefficiently early (Bayer, 2007; Kamoto and Okawa, 2014). Building on these findings, the aim of this paper is to design a contract that combines aspects of cooperative and non-cooperative bargaining to overcome the specific problem that, on one hand, economically efficient investment can be achieved only by cooperative bargaining, while on the other hand non-temporal and non-cooperative bargaining assigns all the profit to the first mover, that is, the supply chain dominant firm. We show that such a contract design exists in a dynamic continuous-time setting and that the design has the added advantage of allowing the supply chain dominant firm to keep information private. As a result, our paper also contributes to the literature on principal-agent theory with one-sided incomplete information, i.e. when the principal has private information (see e.g. Myerson (199), Maskin and Tirole (1990, 1992)). In particular, our results fortify the findings of the traditional literature dealing with the informed-principal problem insofar as the principal, i.e. the dominant firm, does not perform worse than in the full-information case when retaining some discretion. The remainder of the paper is structured as follows. Section 2 sets out the general assumption and model structure. The analytical solutions for the extreme poles of negotiating a contract, that is, either completely non-cooperative or completely cooperative bargaining, are presented in Subsections 2.1 and 2.2 and serve as benchmarks for the later analysis. Subsection 2.3 defines the requirements for an optimal non-cooperative solution and Section 2.4 gives an example of such. Section 3 concludes and offers some suggestions for future research. 2. Preliminaries We consider a (potential) supply chain consisting of the multinational key company K and a local retailer R. The key-company produces a good with manufacturing costs of cK N 0 per unit and has established a binding retail price of pR N cK per unit. In t0 the key company has the opportunity to invest in the expansion of its business into a new foreign market. There, it depends on the cooperation of the local retailer R. The expansion requires the key company to make an investment of IK N 0. Likewise, the retailer has to make an investment of IR N 0.1 The (potential) revenues v(t) in the local market evolve stochastically over time due to overall economic uncertainty as well as due to uncertainties regarding customer's attitudes and behavior. In particular, we assume that they follow the geometricBrownian motion dvðt Þ ¼ μvðt Þdt þ σvðt ÞdW ðt Þ; vð0Þ ¼ v0 N0
ð1Þ
revenues, μR as its growth rate, and dW(t) with v0 N 0 as the (potential) revenues at time t0, σ≥ 0 as the volatility of the (potential) pffiffiffiffiffi as the increment of a Wiener process with zero mean and variance equal to dt which is correlated to the market portfolio. The degree of this correlation is assumed to be constant and measured by ρ ∈ (−1, 1). After the expansion in the local market, the keycompany's profit at time t ≥t0 equals pKp−cK vðtÞ. Here, pK ∈ [cK, pR] denotes the sales price to the retailer. Likewise, the retailers profit R at time t≥ t0 equals pR −ppK −cR vðtÞ with cR ∈ [0, pR − pK − cK] as the retailer's operational costs per unit sold. R
1 For example, the key-company has to bear costs for expanding its production and logistics network, while the retailer has to bear costs for preparing the distribution network.
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We assume that v0 , μ and σ derivate from market research done at t0 and hence are known by both companies.2 However, after t0 and before the expansion takes place only the retailer is able to estimate the current (potential) revenues in the local market. After the expansion, the exact revenues are directly observable by the retailer, while they are still unobservable for the keycompany, which at most may observe the orders of the retailer. Both companies are risk-averse and base their calculations on the CAPM formula (see Merton (1973)), with r ≥μ −λρσ as the riskless interest rate and λR as the market price of risk which is assumed to be constant. Thus, the expected present value of the vðtÞ key-company's future cash flow stream at time t ≥t0 equals pKp−cK rþλρσ −μ , which we denote as the key-company's present value. R vðtÞ pR −pK −cR : Likewise, the retailer's present value equals p rþλρσ−μ R
2.1. Efficient investment timing: the cooperative solution Assume that both companies are owned by the same external decision maker. Consequently, the aggregated profit he makes when investing at an arbitrary time, i.e., τ≥ t0 results in: −r ðτ−t 0 Þ
πE ðτÞ ¼ e
pR −cK −cR vðt Þ −I K −IR : pR r þ λρσ −μ
ð2Þ
Following Dixit and Pindyck (1994), the decision to invest can be regarded as a real option that should be exercised at the efficient investment time τE⁎ = inf{t≥ t0 | v(t) ≥vE⁎}, which is determined by the efficient investment threshold vE ¼
β ðr þ λρσ −μ ÞðIK þ IR ÞpR ; pR −cK −cR β−1
β¼
1 μ−λρσ þ − 2 σ2
ð3Þ
where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 1 μ−λρσ 2 2r þ 2: − 2 2 σ σ
ð4Þ
Following this rule, the expected profit of the external decision maker equals 8 β ðpR −cK −cR ÞvE v0 > > −ðIK þ IR Þ < p ð r þ λρσ −μ Þ vE R GE ¼ ð p −c −c Þv > R K R 0 > : −ðI K þ IR Þ pR ðr þ λρσ−μ Þ
vE Nv0
ð5Þ
vE ≤v0
2.2. Taking full advantage of being the first mover: the non-cooperative solution Now, we assume a sequential bargaining process that, unlike cooperative bargaining, allows the key company to keep information about cK and IK confidential.3 First, at time t0 the key company offers a sales price pK N 0 to the retailer. Then, the retailer immediately decides whether it will accept or reject the offer. In such a static ultimatum game setting the key company gains πU;K ðpK Þ ¼
ðpK −cK Þv0 −I pR ðr þ λρσ −μ Þ K
ð6Þ
with the expansion of its business, while the retailer gains πU;R ðpK Þ ¼
ðpR −pK −cR Þv0 −I : pR ðr þ λρσ−μ Þ R
ð7Þ
Hence, the key company could make an offer pK,U⁎ that is the highest the retailer could accept, i.e. πU,R(pK,U⁎) = 0, and that simultaneously would not represent a loss to the key company itself, i.e., πU, K(pK,U⁎) ≥ 0. Thus, the parties could possibly agree on the þI R ÞpR ðr þ λρσ −μÞ ¼: vU : Consequently, the optimal offered sales price would bepK;U ¼ pR −cR − terms of the deal if v0 ≥ ðpðIK−c K −cR Þ R I R pR ðrþλρσ−μÞ : However, it is well known that in such a setting the key company receives the entire surplus whereas the retailer v0 receives nothing. Thus, the retailer is indifferent between accepting the offer and rejecting it. According to Güth et al. (1982) the key-company should therefore at least leave the minimal positive amount to the retailer. Besides that, the key company 2 Any potential costs for doing this market research would be sunk costs as they have already have been borne by the key-company before t0. Consequently, they can be ignored in our model. 3 In particular, the retailer is not able to verify the exact values of cK and IK from the offered sales price pK,U⁎.
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may wish to grant a little profit of x N 0 to its retailer to keep the retailer loyal and economically viable. Consequently, the offered −μÞx and hence becomes pK;U ¼ pR −cR − ðIR −xÞpR ðrþλρσ−μÞ . For the total surplus in the ultimatum sales price decreases by pR ðrþλρσ v0 v0 game we obtain ðp −c −c Þv GU ¼ max 0; R R K 0 −I K −IR : pR ðr þ λρσ−μ Þ
ð8Þ
Hence, as long as vE⁎ N v0 we get that GU b GE. This overall loss is due to losing the value of being able to defer the investment. Consequently, in the ultimatum game, investment usually occurs inefficiently early.
2.3. Requirements on an optimal contractual solution As we have seen in the previous sub sections from the key-company's perspective cooperative bargaining as well as non-cooperative bargaining via the ultimatum game each comes with certain advantages and disadvantages. While the outcome of cooperative bargaining is Pareto-optimal, i.e. the total surplus is maximized due to efficient investment timing, the ultimatum game allows the key-company to keep the total surplus and to keep its information on cK and IK private. Consequently, the question arises if a contractual solution exists that combines the described advantages of the key-company after cooperative bargaining and non-cooperative bargaining. If vE⁎ ≤ v0 it is obvious that the ultimatum game already leads to the optimal outcome for the key company because it maximizes the total surplus (an external decision maker would also invest at t0) and grants it entirely to the key company. Furthermore, the key-company can keep its information on cK and IK private. Hence, in this case the keycompany does not need a special contract to improve its situation. In the following we, therefore, assume that vE⁎ N v0. In particular, an optimal contractual solution has to meet the following four requirements: (i) (ii) (iii) (iv)
The total surplus of the deal is maximized, i.e. investment timing is efficient. The key-company can decide alone on the division of the surplus, i.e. it can choose the expected profit x of its retailer. The key-company can keep its information on cK and IK private. The contractual solution does not presuppose that the revenues at the time of investment or at any later point in time are verifiable by the key-company or that both parties could make any contracts on their basis.
While the first requirement is self-explanatory the second requirement not only enables the key-company to keep the entire surplus but also allows him to voluntarily grant the retailer a part of this profit. Reasons for providing such a voluntarily grant might, e.g., be justified in order to perpetuate the loyalty of the retailer, to secure the economic viability of the retailer, or to improve the image of the company towards its current retailers as well as to possible future retailers. Advantageously, the contract proposed in this article does not provide an economic rule to decide on an optimal grant to the retailer. Rather it allows to exogenously optimizing for it as the resulting investment time in the proposed contract does not depend on the amount of this grant. The third requirement guarantees that the retailer does not get to know the exact values of cK and IK. This possibility to keep some information private may be advantageous to the key-company.4 For example, due to strategic considerations or because the key-company has committed to a third party not to share this information (Li and Zhang, 2008), or as without knowing the exact values of cK and IK the retailer is not able to determine the key-company's profit of the investment.5 The fourth requirement results from the situation that the realized revenues in the local market are not verifiable by the key-company and thus cannot be part of the contractual solution.
2.4. An optimal contractual solution In the following proposition we provide an optimal contractual solution, i.e. a solution that meets all the four requirements described above. Thereby, we build on the approaches of Lambrecht (2004) and Lukas and Welling (2012) who both describe a non-cooperative option game in which one party (the first-mover) offers the other party (the second-mover) a certain payment and then the second-mover has the right to decide on the investment-timing. However, if the first-mover simply offers an equity stake, i.e. percentage participation on the realized revenue, the second-mover will time the investment inefficiently (at least from the first-movers perspective). Likewise, offering a fixed payment does not lead to efficient investment timing, either. Surprisingly, an ideal combination of an “equity” stake and a fixed payment exists that leads to efficient investment timing and simultaneously meets the other two requirements described above:
4 Obviously, the contractual solution of this article does not prohibit the key-company to share all of its private information, but gives the key-company some flexibility regarding the information sharing, without jeopardizing the social-efficiency of the negotiation. 5 Consequently, the retailer cannot compare the key-company's profits with his own profits and therefore is not able to measure the fairness of the deal.
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2.4.1. Proposition: It is optimal to the key-company to let the retailer decide on the investment time τ ≥t0 and to offer him a sales price of
pK;C ðxÞ ¼ pR −cR −ðβ−1Þ
pR −cK −cR v0 −β x ðIR þ IK Þ vE
ð9Þ
in combination with a fixed payment at time τ of −β v x: ψK;C ðxÞ ¼ IR þ ð1−βÞ 0 vE
ð10Þ
2.4.2. Proof From the retailer's perspective the right to time the investment has to be regarded as a real option of perpetual American type. Exercising the option at time τ the retailer gains. 0 1 p −c −c −β ðβ−1Þ R K R pR − pK;C ðxÞ−cR ðIR þ IK Þ B C v vðτÞ þ ψK;C ðxÞ−IR ¼ @ vðτÞ þ ð1−βÞA 0 x πR ðτ; xÞ ¼ pR ðr þ λρσ−μ Þ pR ðr þ λρσ−μ Þ vE −β x v0 ¼ βvðτÞ−ðβ−1ÞvE vE vE
ð11Þ
Hence, the retailer will choose the investment time τC⁎ that maximizes its expected profit discounted on time t0. Thus, τC⁎ is given by −β −β v0 x v0 x −rðτC −t0 Þ −r ðτC −t 0 Þ −r ðτ−t 0 Þ ¼ max E e E βv τC −ðβ−1ÞvE e βvðτ Þ−ðβ−1ÞvE e vE τ∈½t 0 ;∞ vE vE vE
ð12Þ
From Dixit and Pindyck (1994) we know that it is optimal for the retailer to invest as soon as the value of the traded asset reaches a certain threshold vC⁎, i.e.
τC ¼ inf t ≥t 0 jvðt Þ≥vC ;
ð13Þ
and that the value of the retailer's option to invest can be calculated by f R vC ¼
−β β v0 x v0 : βvC −ðβ−1ÞvE vE vC vE
ð14Þ
Due to
∂f R ðvC Þ ¼ ∂vC
−β β β −β β v0 x v0 v0 v x v0 ððβ−1ÞÞ β − βvC −ðβ−1ÞvE ¼ − 0 vC −vE β β vE vC vE vC vE vC vC vE vC
ð15Þ
we can easily see that vC⁎ = vE⁎ is the unique solution of the equation
∂f R ðvC Þ ¼ 0: ∂vC
ð16Þ
Thus, the retailer will exercise the option at the efficient investment time. Considering that vC⁎ = vE⁎ we get from Eq. (14) that f R vC ¼
−β β v0 x v0 ¼ x: βvC −ðβ−1ÞvE vE vC vE
ð17Þ
Consequently, the key-company can solely decide to offer a certain profit with expected present value x N 0 to the retailer.6 Furthermore, the incentive contract has the advantage of keeping information private. By “private” information we mean that it is not possible for the retailer to discover the key company's strategic variables from the offered premium. As vE⁎ is known, the retailer can according to Eq. (3) determine a linear relationship between cK and IK. However, the size of vE⁎ cannot be used to gain full 6 In this regard, it should be noted that for x=0 the retailer would be indifferent between investing at any point in time and never investing, hence the key-company should - in analogy to the ultimatum game - at least offer the minimal positive amount x to the retailer.
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information regarding the key company's true investment cost IK nor its true size of manufacturing costs cK. In particular, for every z N 0 we get that |{(IK,cK)|vE⁎(IK,cK) = z}| = ∞. Finally, in the contractual solution (see Eqs. (9) and (10)) the key-company does not offer anything that would depend on the revenue at τ or at a later point in time □ In the ultimatum game the retailer has to decide on a take-it-or-leave-it base. In contrast, the contractual solution described above seemingly provides some flexibility to the retailer, i.e. he can wait with the decision and choose the individually optimal investment time. Though, the contract is designed in a way that (except of a voluntarily payment) only the key-company generates a surplus with the investment. Hence, it is the key-company and not the retailer who profits from the arising flexibility value. This result is achieved by offering a cost-plus contract (see for example Agrell et al., 2004, p. 6) where on one hand the retailer is investment-time-independently compensated with his investment costs IR and operating costs cR and on the other hand an additional voluntarily payment is designed in a way that its expected present value is maximized when the investment takes place at the socially-efficient investment time τE⁎. 2.5. The benefit of the optimal contractual solution In the following we analyze under which circumstances the contractual solution generates the highest benefit compared to the ultimatum game. Therefore, we first calculate this benefit, i.e. the difference of total gains from the contractual solution and the total gains from the ultimatum game: 8 β ðpR −cK −cR ÞvE v0 > > > −ðI K þ IR Þ > > p ð r þ λρσ−μ Þ v > > R E β > > v0 < ðpR −cK −cR ÞvE −ðI K þ IR Þ GE −GU ¼ pR ðrþ λρσ−μ Þ vE > > > ð Þv p −c −c > R R K 0 > −I −I − > > > pR ðr þ λρσ−μ Þ K R > : 0
v0 ≤vU
vU bv0 bvE
:
ð18Þ
vE ≤v0
In particular, we have to distinguish three cases. First, if the outcome of the contractual solution is not ultimate investment and the outcome of the ultimatum game would be no investment, i.e. for v0 ≤ vU⁎, the ultimatum gain generates no profit. Consequently, the benefit of the contractual solution compared to the ultimatum game equals the central planner's value of the option to invest. As can be easily seen from Eq. (18) this option value is increasing with increasing v0. Second, if the outcome of the contractual solution is not ultimate investment and the outcome of the ultimatum game would be investment, i.e. for vU⁎ b v0 b vE⁎, the benefit of the contractual solution compared to the ultimatum game is the difference of the central planner's value of the option to invest and the intrinsic value of this option. That is the extrinsic value of this option. However, as the total gain of the ultimatum game
Fig. 1. The additional value GE −GU of the contractual solution compared to the ultimatum game in dependence of the initial value v0 of the potential revenues and for different values of uncertainty and correlation to the market portfolio (fat line (benchmark case): σ=0.2,ρ=0; solid line (higher uncertainty): σ=0.3,ρ=0; dash-dotted line (lower uncertainty): σ= 0.1, ρ = 0; dashed line (negative correlation): σ = 0.2, ρ = − 0.2; dotted line (positive correlation): σ= 0.2, ρ = 0.2). Other parameter values: r=0.1,μ=0.05,λ=0.2,IK =10,IR =10, cK =1,cR =1,pR =4.
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increases faster in v0 than the central planner's option value (see Eq. (18)) the benefit of the contractual solution compared to the ultimatum game is decreasing in v0. Third, if the outcome of the contractual solution is ultimate investment, i.e. for vE⁎ ≤v0, its total gain equals the total gain of the ultimatum game which due to vE⁎ N vU⁎ leads to ultimate investment as well. Consequently, the benefit of the contractual solution compared to the ultimatum game is zero. Finally, we can therefore conclude that the benefit of the contractual solution compared to the ultimatum game is the highest, if the investment breaks even in the ultimatum game, i.e. if v0 =vU⁎ (see also Fig. 1). Fig. 1 furthermore depicts that the benefit GE − GU of the contractual solution in comparison to the ultimatum game increases with the uncertainty σ about the value of the traded asset. This is due to the flexibility value of the contractual solution that increases with uncertainty while the profit of the ultimatum game is not influenced by the uncertainty as long as the traded asset is uncorrelated to the market portfolio. However, a positive [negative] correlation of the traded asset and the market portfolio leads to a lower [higher] break-even point vU⁎ and a lower [higher] benefit of the contractual solution compared to the ultimatum game. Finally, another important result is that retaining information does not put the dominant firm in a worsen position compared to the full-information case. Rather, the outcome of an asymmetric Nash-bargaining game will assign less surplus assigned to the dominant firm than in the one-sided incomplete information case. To see this, we have to compare the full-information case, which is represented by Eq. (5) with the surplus generated by means of the optimal contract. Obviously, both strategies yield the same surplus with the only difference that the bargaining power determines how much surplus the dominant firm receives from the joint investment. Fig. 2 depicts the influence of several other parameters on the benefit of the contractual solution compared to the ultimatum game. In particular, a higher riskless interest rate leads to a higher break-even point vU⁎ and a lower benefit of the contractual solution compared to the ultimatum game. Contrarily, a higher growth rate of the revenues leads to a lower break-even point vU⁎ and a higher benefit of the contractual solution compared to the ultimatum game. A higher retail price also leads to a lower break-even point, but does hardly influence the benefit of the contractual solution. Finally, higher investment costs lead to a higher break-even point and come with a greater benefit of the contractual solution. 3. Conclusion Negotiating contracts is at the heart of every supply chain design. If structured cooperatively each party has to make its information public in order to jointly maximize profits and equitable profit sharing. If, however, the negotiations are structured noncooperatively profit sharing usually is assumed to be biased in favor of the first mover but inefficiencies arise due to postponement of strategic investment decisions. A natural question arises as to whether it is possible to design a contract that combines the positive aspects of both types of negotiating, i.e. cooperative and non-cooperative. In this paper, we show that such is indeed possible and when drawn up properly, such a contract provides an incentive to not further delay investment. Moreover, the
Fig. 2. The additional value GE −GU of the contractual solution compared to the ultimatum game in dependence of the initial value v0 of the traded asset and for different parameter values (fat line (benchmark case); solid line (higher retail price): pR =5; dash-dotted line (higher interest rate): r=0.12; dashed line (higher growth rate): μ=0.07; dotted line (higher investment costs): IK =15). Benchmark parameter values: σ=0.2,ρ=0,r=0.1,μ=0.05,λ=0.2,IK =10,IR =10, cK = 1,cR =1,pR =4.
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