Nakamura Yasuhiko

English

Nakamura, Yasuhiko

48647

Waseda University Repository

  21COE-GLOPE Working Paper Series                                                              If you have any comment or question on the working paper series, please contact each author. When making a copy or reproduction of the content, please contact us in advance to request permission. The source should explicitly be credited. GLOPE Web Site: http://www.waseda.jp/prj-GLOPE/en/index.html Bargaining over Managerial Contracts in Delegation Games: The Differentiated Goods Case    Yasuhiko Nakamura          Working Paper No. 40 1 IntroductionThis paper presents a theoretical analysis of bargaining on managerial delegation between owner-shareholders and managers in a differentiated-products market. Modern corporate governancecodes includes clauses retiring the disclosure of managerial compensation in order to protectowner-shareholders against the opportunistic behavior of managers. This paper aims to ex-plain the effects of managerial power and the disclosure of managerial compensation on marketoutcomes in the case wherein each firm produces a differentiated good.A theoretical analysis of the separation of ownership and management challenges the va-lidity of the traditional assumption of a firm as the sole profit maximizing agent. With thedevelopment of a principal-agent model, studies have been conducted to clarify the interactionbetween owner-manager contracts and market behavior in the context of an oligopolistic indus-try with managerial delegation. In their well-known articles, Fershtman and Judd (1987) andSklivas (1987) revealed that profit-maximizing owners sometimes choose to divert the preferenceof their managers away from profit maximization. They investigated a two-stage duopolisticgame where in the first stage, each profit-maximizing owner chooses a compensation scheme forhis or her manager that is a linear combination of profit and sales. Based on the two semi-nal works, the literature on delegation games extended its scope to alternative compensationschemes. Salas Fumas (1992) and Miller and Pazgal (2002) considered the compensation schemeas a linear combination of the profit of a firm and its rival. Jansen et al. (2007) investigated thecase in which each manager’s remuneration is a weighted sum of the firm’s profit and marketshare.Another major branch of the literature on managerial delegation considers bargaining is-sues between an owner and a manager in various environments. Since the pioneering work byFershtman (1985), this type of model has particularly focused on wage bargaining. Szyman-ski (1994) and Bughin (1995) analyzed the influence of the delegation of wage bargaining withunions to managers on product market competition, depending on the type of the managerialcompensation scheme. Along the lines of these works, Witteloostuijn et al. (2007) consideredthe bargaining between owners and managers over their compensation contracts, in order toexplain the disclosure obligation that is central to many modern corporate governance codes.More precisely, they analyzed the two-stage duopolistic game where in the first stage ownersnegotiate with their managers over managerial delegation contracts. Subsequently, in the secondstage, Cournot competition with a homogeneous good is observed.The main purpose of this paper is to expand the scope of the results of Witteloostuijn et al.(2007) with respect to the sales delegation case to those of a differentiated-products duopolisticgame. In this paper, the game operates in the same manner as that in Witteloostuijn et al.(2007). However, we investigate both Cournot and Bertrand competitions. In short, we extendthe scope of the strategic managerial delegation model with bargaining between owners andmanagers by considering both product differentiation and the change in firms’ strategic variables.Through these analyses, we intend to determine whether or not the results of Witteloostuijn et al.(2007) of the sales delegation is robust against a change(s) in the degree of product differentiationand/or the firms’ strategic variables. We obtain the result that competition is promoted inboth Cournot and Bertrand fashions irrespective of the degree of product differentiation, as1the relative bargaining power of the manager increases. Thus, in the context of a symmetricduopolistic competition with bargaining between owners and managers, we find that the resultsof Cournot competition with homogeneous goods are robust against the changes in the degreeof product differentiation and firms’ (managers’) strategic variables.The remainder of this paper is organized as follows. In Section 2, we formulate the basicsetting of the delegation game of the two types of models considered in this paper. In Section3, we consider the case of quantity competition. In Section 4, we investigate the case of pricecompetition. Section 5 presents the concluding remarks.2 ModelWe consider a symmetric duopolistic model using a standard product differentiation model asin Singh and Vives (1984). A representative consumer’s utility is denoted asU(q1, q2) = a (q1 + q2)− 12[(q1)2 + 2bq1q2 + (q2)2]+ q,where qi represents the quantity of good i (= 1, 2), and b ∈ (0, 1) represents the degree ofproduct differentiation. Note that q denotes a numeraire good. The utility function generatesthe system of linear demand functions:qi =a (1− b)− pi + bpj1− b2 , b ∈ (0, 1) , i, j = 1, 2, i 6= j.Then, the inverse demand functions can be inverted to obtainpi = a− qi − bqj , b ∈ (0, 1) , i, j = 1, 2, i 6= j,where pi represents the price of good i (= 1, 2). We assume that both the firms have constantmarginal costs of production, c (< a). The profit of firm i is given byΠi = (a− qi − bqj − c)qi=a (1− b)− pi + bpj1− b2 (pi − c) , i, j = 1, 2, i 6= j.Usually, social welfare, denoted by W , is measured as the sum of consumer surplus (CS) andproducer surplus (PS):W = CS + PS,where PS = Π1 +Π2, and CS is given byCS =12[(q1)2 + 2bq1q2 + (q2)2],=2a2 (1− b) + (p1)2 − 2bp1p2 + (p2)2 − 2a (1− b) (p1 + p2)2 (1− b2) .We consider the situation in which a firm’s owner decides to delegate the control of his orher assets to a manager, similar to the case in Fershtman and Judd (1987) and Sklivas (1987).Both the firms’ owners can assess the performance of their managers according to two readily2observable indicators–the output and profit of the firm. In this case, the objective of eachmanager is given byUi = Πi + θiqi, θi ∈ R, i = 1, 2,where parameter θi measures the relevance of sales. The manager of firm i can maximize his orher payoff by choosing the output qi or the price pi that maximizes Ui (i = 1, 2). This can besupported by the assumption that the payoff to the manager of firm i is represented as λi+µiVifor some real number λi and some positive number µi (i = 1, 2). Similar to many existing works,we assume that the payoffs to the managers are negligible as compared to profits, because weemphasize the impact of managerial delegation on the equilibrium outcomes.We propose the following two-stage delegation game. In the first stage, an owner and his orher manager bargain over the incentive parameter θi in each firm (i = 1, 2). Subsequently, inthe second stage, each manager simultaneously decides his or her output or price while beingaware of the level of his or her own incentive parameter. Thus, owners and managers bargainover their incentive parameters in the sales delegation contracts such that both their profits andpayoffs are enhanced through their collusive market behaviors.The bargaining in the first stage is generally formulated in the same manner as in Binmoreet al. (1986). For each firm, the equilibrium outcome of the bargaining process coincides withthe Nash bargaining solution of the following equation in terms of the incentive parameter θi:Bi = Uβi ·Π1−βi , i = 1, 2,where β ∈ [0, 1) denotes the measure for the relative bargaining power of the manager. Notethat in this paper, we restrict our attention to the symmetric case in which all managers have thesame bargaining powers β. As a result, we assume that owners select sufficiently homogeneousmanagers both in terms of their bargaining powers and the preferences to their payoffs. Thedisagreement point of both owner and manager is zero: If the bargaining process breaks down,the managers are unable to procure the other jobs, and the owners are unable to operate runtheir firms since they do not have adequate time or managerial skills.3 Quantity CompetitionIn this section, we consider how profits, consumer surplus, and social welfare depend on therelative bargaining power of the manager, β, for the sales delegation in the case of quantitycompetition in a differentiated-products duopolistic setting.As usual, we begin by solving the second stage. Given a pair of incentive parameters θ1 andθ2, each manager maximizes his or her objective Ui with respect to qi (i = 1, 2). The conditionfor each manager is given by∂Ui∂qi= a− c− 2qi − bqj + θi = 0, i, j = 1, 2, i 6= j.Solving the system of the above equations for q0 and q1, we obtain the output of each firm asfollows:qi =a (2− b)− c (2− b) + 2θi − bθj4− b2 , i, j = 1, 2, i 6= j.3According to the first-order condition of the above system, we haveUi = (qi)2 , Πi = (qi)2 − θiqi, i = 1, 2.Thus, we obtain the following results:∂Ui∂θi=4qi4− b2 > 0,∂2Πi∂θ2i= −4(2− b2)(4− b2)2 < 0, ∀b ∈ (0, 1) , i = 1, 2.In short, similar to the results of the sales delegation case of Witteloostuijn et al. (2007), weobtain the result that the objective function of the manager Ui is an increasing function of θi,and the profit of each firm is an concave function of θi (i = 1, 2). The interests of the ownerand manager of each firm are different. Therefore, it is appropriate for them to bargain over theincentive parameter, θi (i = 1, 2).We now proceed to the analysis of the first stage of the game. Considering the outcomes inthe first stage, we rewrite the Nash product for the bargaining problem of each firm as follows:Bi = Uβi ·Π1−βi = (qi)2β ·[(qi)2 − θiqi]1−β, i = 1, 2.Thus, analogous to Witteloostuijn et al. (2007), we obtain the following first-order condition forthe bargaining of each firm:∂Bi∂θi= 0⇔ [2qi − θi (1 + β)] ∂qi∂θi− qi (1− β) = 0, i = 1, 2.Taking into account that ∂qi/∂θi = 2/(4− b2) and that the problem’s symmetry θ = θ1 = θ2,we obtain∂Bi∂θi= 0⇔ a[b2 (1− β) + 4β]− b2 (1− β) (c− θ)− 2bθ (1 + β)− 4 (cβ + θ)(2− b) (2 + b)2 = 0,yieldingθ∗ = θ∗i =b2 (1− β) + 4β4 + 2b (1 + β)− b2 (1− β) (a− c) , i = 1, 2.The equilibrium outcomes (denoted by superscript “∗”) are obtained as follows:q∗i =2 (1 + β)4 + 2b (1 + β)− b2 (1− β) (a− c) ; p∗i =a(2− b2) (1− β) + 2c (1 + b) (1 + β)4 + 2b (1 + β)− b2 (1− β) ;Π∗i =2(2− b2) (1− β) (1 + β)[4 + 2b (1 + β)− b2 (1− β)]2 (a− c)2 ; i = 1, 2;CS∗ =4 (1 + b) (1 + β)2[4 + 2b (1 + β)− b2 (1− β)]2 (a− c)2 ;W ∗ =4 (1 + β)[3− β + b (1 + β)− b2 (1− β)][4 + 2b (1 + β)− b2 (1− β)]2 (a− c)2 .Furthermore, we obtain the following result:∂Π∗i∂β= −4(2− b2) [4β + 2b (1 + β) + b2 (1− β)][4 + 2b (1 + β)− b2 (1− β)]3 (a− c)2 < 0,4∀b ∈ (0, 1) , ∀β ∈ [0, 1) , i = 1, 2.In short, the partial derivative of the equilibrium profit of each firm with β is negative for allthe values of b ∈ (0, 1) and β ∈ [0, 1). On the other hand, we have∂CS∗∂β=16 (1 + b)(2− b2) (1 + β)[4 + 2b (1 + β)− b2 (1− β)]3 (a− c)2 > 0, ∀b ∈ (0, 1) , ∀β ∈ [0, 1) ,and∂W ∗∂β=8(2− b2)2 (1− β)[4 + 2b (1 + β)− b2 (1− β)]3 (a− c)2 > 0, ∀b ∈ (0, 1) , ∀β ∈ [0, 1) .With respect to the equilibrium consumer surplus and social welfare, the partial derivatives withβ are positive for all the values of b ∈ (0, 1) and β ∈ [0, 1). The preceding results are summarizedin Proposition 1.Proposition 1. If each owner in a symmetric Cournot duopoly hires a manager who receives apayoff through the managerial incentive contract based on profits and sales, and if the incentiveparameter on sales is the outcome of a bargaining process, then the corresponding delegationgame has a unique subgame perfect Nash equilibrium, which depends on the bargaining power βof the managers. Moreover, if the bargaining power of the managers increases, the equilibriumprofit of each firm decreases, whereas the consumer surplus and social welfare increases.Note that similar to Witteloostuijn et al. (2007), as the measure for the relative bargainingpower of the manager β approaches 1, the equilibrium profit of each firm, Π∗i approaches 0,whereas the consumer surplus, CS∗ and social welfare, W ∗ gradually increase.4 Price CompetitionIn this section, we analyze the price competition case in which each firm produces a differentiatedgood. We again use backward induction and consider a two-stage process in which the ownersand managers bargain over the incentive parameter in the first stage, and both the firms competewith regard to prices in the second stage.Given a pair of incentive parameters θ1 and θ2, each manager maximizes the objective func-tion Ui with respect to pi (i = 1, 2). The first-order condition for each manager is given by∂Ui∂pi=a (1− b) + c− 2pi + bpj − θi1− b2 = 0, i, j = 1, 2, i 6= j.Solving the system of the above equations for p0 and p1, we obtain the price of each firm asfollows:pi =a(2− b− b2)+ c (2 + b)− 2θi − bθj4− b2 , i, j = 1, 2, i 6= j.Considering the first-order condition of each manager, we obtainUi =[a (1− b)− pi + bpj ]21− b2 ,5Πi =[a (1− b)− pi + bpj ] [a (1− b)− pi + bpj − θi]1− b2 , i = 1, 2.Thus, we obtain the following results:∂Ui∂θi=2(2− b2)(4− b2) qi > 0,∂2Πi∂θ2i= − 4(2− b2)(4− b2)2 (1− b2) < 0, ∀b ∈ (0, 1) , i = 1, 2.Therefore, similar to quantity competition, we find that in price competition the objectivefunction of each manager Ui is an increasing function of θi, and the profit of each firm is aconcave function of θi (i = 1, 2).Next, we analyze the bargaining between owners and managers in the first stage. In the caseof price competition, the bargaining problem is as follows:Bi = Uβi ·Π1−βi={[a (1− b)− pi + bpj ]21− b2}β·{[a (1− b)− pi + bpj ] [a (1− b)− pi + bpj − θi]1− b2}1−β,i = 1, 2.Considering the problem’s symmetry, θ = θ1 = θ2, we obtain∂Bi∂θi= 0⇔ a (1− b) [4β − b2 (1 + β)]− b3c (1 + β) + b2 (1 + β) (c+ θ)− 4 (cβ + θ) + 2b [2cβ + θ (1− β)] = 0, i = 1, 2,yieldingθ∗ = θ∗i =(1− b) [4β − b2 (1 + β)]4− 2b (1− β)− b2 (1 + β) (a− c) , i = 1, 2.We obtain the equilibrium outcomes as follows:p∗i =2a (1− b− β + bβ) + c (2− b2) (1 + β)4− 2b (1− β)− b2 (1 + β) , q∗i =(2− b2) (1 + β)(1 + b) [4− 2b (1− β)− b2 (1 + β)] (a− c) ;Π∗i =2(2− 2b− b2 + b3) (1− β2)(1 + b) [4− 2b (1− β)− b2 (1 + β)]2 (a− c)2 ; i = 1, 2;CS∗ =(2− b2)2 (1 + β)2(1 + b) [4− 2b (1− β)− b2 (1 + β)]2 (a− c)2 ;W ∗ =(2− b2) (1 + β) [2 (3− β)− 4b (1− β)− b2 (1 + β)](1 + b) [4− 2b (1− β)− b2 (1 + β)]2 (a− c)2 .Furthermore, we obtain the following result:∂Π∗i∂β= −4(2− 2b− b2 + b3) [2b (1− β) + 4β − b2 (1 + β)](1 + b) [4− 2b (1− β)− b2 (1 + β)]3 (a− c)2 < 0,∀b ∈ (0, 1) , ∀β ∈ [0, 1) .6Thus, we find that the equilibrium profit of each firm decreases for all the values of b ∈ (0, 1)and β ∈ [0, 1) as the relative bargaining power of the manager, β, increases. On the other hand,we obtain∂CS∗∂β=8 (1− b) (2− b2)2 (1 + β)(1 + b) [4− 2b (1− β)− b2 (1 + β)]3 (a− c)2 > 0, ∀b ∈ (0, 1) , ∀β ∈ [0, 1) ,and∂W ∗∂β=16 (1− b)2 (2− b2) (1− β)(1 + b) [4− 2b (1− β)− b2 (1 + β)]3 (a− c)2 > 0, ∀b ∈ (0, 1) , ∀β ∈ [0, 1) .Thus, we find that both the equilibrium consumer surplus and social welfare improves as therelative bargaining power of the managers, β, increases in contrast to the equilibrium profit ofeach firm. The preceding results are summarized in Proposition 2.Proposition 2. If each owner in a symmetric Bertrand duopoly hires a manager who receives apayoff through the managerial delegation contract based on profits and sales, and if the incentiveparameter on sales is the outcome of a bargaining process, then the corresponding delegationgame has a unique subgame perfect Nash equilibrium, which depends on the bargaining power βof the managers. Moreover, if the bargaining power of the managers increases, the equilibriumprofit of each firm decreases, whereas the consumer surplus and social welfare increase.Note that analogous to quantity competition, as β approaches 1, the equilibrium profit ofeach firm, Π∗i approaches 0, whereas the consumer surplus, CS∗ and social welfare,W ∗ graduallyincrease.5 ConclusionThis paper examined the bargaining between owners and managers over managerial incentivecontracts for the sales delegation case in two types of differentiated-products markets withCournot and Bertrand competitions. We analyzed whether or not the results of Witteloostuijnet al. (2007) are robust against the changes in the degree of product differentiation and/orfirms’ (managers’) strategic variables. We obtained the result that the managerial power of themanagers, β, is positively associated with the equilibrium consumer surplus and social welfare,similar ot that in Witteloostuijn et al. (2007). Therefore, the result of the managerial delegationgame with bargaining between owners and managers is robust against such extensions.Similar to most existing works, we assume the simultaneous-movement of both the firmsin the market stage. The interesting extension of our model is to examine whether or notthe results of this paper are robust in the case of the sequential-movement of both the firms.Futuere research should focus on other compensation schemes, e.g., the relative profit delegationand market share delegation cases, and should confirm the robustness of the results in this paper.ReferencesBinmore, K., A. Rubinstein and A. Wolinsky (1986) “The nash bargaining solution ineconomic modelling” Rand Journal of Economics 17, 176–188.7Bughin, J. (1995) “Unions and strategic managerial incentives” Economics Letters 47, 95–100.Fershtman, C. (1985) “Managerial incentives as a strategic variable in duopolistic environ-ment” International Journal of Industrial Organization 3, 245–253.Fershtman, C. and K. Judd (1987) “Equilibrium incentives in oligopoly” American EconomicReview 77, 927–940.Jansen, T., A. V. Lier and A. V. Witteloostruijn (2007) “A note on strategic delegation:The market share case” International Journal of Industrial Organization 25, 531–539.Miller, N. and A. Pazgal (2002) “Relative performance as a strategic commitment mecha-nism” Managerial and Decision Economics 23, 51–68.Salas Fumas, V. (1992) “Relative performance evaluation of management” International Jour-nal of Industrial Organization 10, 473–489.Singh, N. and X. Vives (1984) “Price and quantity competition in a differentiated duopoly”Rand Journal of Economics 15, 546–554.Sklivas, S. D. (1987) “The strategic choice of management incentives” Rand Journal of Eco-nomics 18, 452–458.Szymanski, S. (1994) “Strategic delegation with endogenous costs: A duopoly with wagebargaining” International Journal of Industrial Organization 12, 105–116.Witteloostuijn, A, V., T. Jansen and A. V. Lier (2007) “Barganing over managerialcontracts in delegation games: Managerial power, contract disclosure and cartel behavior”Managerial and Decision Economics 28, 897–904.8

Bargaining over Managerial Contracts in Delegation Games: The Differentiated Goods Case

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Bargaining over Managerial Contracts in Delegation Games: The Differentiated Goods Case

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