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1 DISCUSSION PAPER SERIES IN ECONOMICS AND MANAGEMENT Strategic Incentives for Managers in Contests Matthias Kräkel Discussion Paper No GERMAN ECONOMIC ASSOCIATION OF BUSINESS ADMINISTRATION - GEABA

2 Strategic Incentives for Managers in Contests Matthias Kräkel, University of Bonn Abstract Owners usually induce their managers to maximize profits instead of sales. This paper shows that in the context of strategic interactions between managers on markets which can be characterized as contests, owners may make their managers maximize sales. JEL classification: L1, M2. Key words: contests, delegation, strategic incentives. * I would like to thank Christian Grund and Christine Harbring for helpful comments. ** Prof. Dr. Matthias Kräkel, Department of Economics, BWL II, University of Bonn, Adenauerallee 24-42, D Bonn, Germany, m.kraekel@uni-bonn.de, phone:

3 1 Introduction The separation of ownership from management implies a wide range of agency problems in corporations. While it is well-known in the principal-agent framework that owners should compensate their managers according to profits instead of sales, the same may not hold in a strategic context. Here, it may be advantageous for owners to induce a more aggressive market behavior to their managers by putting a positive weight on sales in the managers compensation schemes. The optimal strategic incentives for managers in oligopolistic markets have already been discussed in some papers, where it is assumed that owners have to decide about a compensation scheme that consists of a linear combination of profits and sales: Under Cournot competition owners put a positive weight on sales, whereas under Bertrand competition it is optimal for owners to put a negative weight on sales. 1 Obviously, the Cournot and the Bertrand model are not the only forms of competition when owners delegate decisions to managers and have to decide about strategic management compensation. This paper combines the strategic incentives approach of Fershtman and Judd 1987 and Sklivas 1987 with contest theory. 2 For this context, the optimal linear combination of profits and sales as strategic incentives for managers is derived. 3 In a lot of situations market competition can be much better characterized by a contest model than by Cournot or Bertrand competition: For example, Dixit 1987, p. 896 mentioned the case of oligopolistic competition for a homogeneous product with unit-elastic demand, where firms indirectly choose different market shares. In addition, we can also think of sit- 1 See Fershtman and Judd 1987, Sklivas For contests or all-pay auctions see, for example, Dixit 1987, Hirshleifer and Riley 1992, pp , Baye et al Delegation in contests has already been discussed by Baik and Kim 1997, Schoonbeek 2000, and Konrad et al. 2000, but not in the Fershtman-Judd-Sklivas framework. 2

4 uations where firms or managers, respectively may end up as contest winners with high sales whereas losing firms only get low sales: There are many cases in which firms must spend some resources in advance to compete for a highly profitable order from a public institution or a private corporation. Since each firm loses its resources spent we can speak of a contest or an all-pay auction. Such situations can be often found in the professional service industries, for example, when advertising firms compete for a given budget of an industrial corporation by elaborating proposals for a new publicity campaign. 2 Model and Results A two-stage model with two risk neutral owners and two risk neutral managers is considered. 4 On the first stage, owner i i =1, 2 has to choose a linear combination O i = α i Π i +1 α i S i of profits Π i and sales S i as incentive scheme for his manager i with α i > 0. 5 On the second stage, the two managers compete in a contest against each other. Here, the managers simultaneously spend resources µ i 0i =1, 2 to become the winner of the contest. These resources are described in monetary terms so that µ i also denotes firm i s costs. 6 There are two possible outcomes for each manager in the contest: Manager i becomes the contest winner and realizes high sales S i = S H for his firm, or he loses and gets only low sales S i = S L <S H. Let p i denote manager i s probability of winning the contest i, j =1, 2; i j. It is assumed that manager i wants to maximize the expected value of O i when choosing µ i whereas owners choose α 1 and α 2 to maximize expected profits EΠ i. 7 In the following, we distinguish two 4 The first stage follows the modeling of Fersthman and Judd 1987 and Sklivas α i 0 is excluded here, because this would result in managers spending infinite resources on the second stage. 6 Such a modeling is standard in contest theory, see the mentioned literature above. 7 As in Fershtman and Judd 1987 there is no disutility of effort for the managers. We can imagine that the compensation for manager i is characterized by A i + B i O i B i > 0 where A i 3

5 different specifications of the winning probability p i = p A i and p i = p B i : 8 1 if µ i >µ j 1 if µ p A i = 1 B if µ 2 i = µ j, p i = 2 i = µ j =0 µ i if µ µ 0 if µ i <µ i +µ j i,µ j 0 j Proposition 1 When p i = p A i then there exist two subgame perfect equilibria where, on the first stage, owner i chooses αi =2and owner j chooses α j = ε with ε being an arbitrarily small but positive number i, j =1, 2; i j and, on the second stage, manager i chooses a mixed strategy according to the cumulative distribution function G i µ =1 α j/α i + α j µ/ S with S = S H S L and µɛ[0, S/α i ], whereas manager j s mixed strategy can be described by G j µ =α i µ/ S with µɛ[0, S/α i ]. 9 Proposition 1 contains some interesting results. First, symmetric equilibria do not exist, although the decision structures of firm 1 and firm 2 are identical. Second, in equilibrium no owner solely induces his manager to maximize profits. Instead of this, there are two asymmetric equilibria where one owner puts a positive weight on profits and a negative weight on sales α i =2, whereas the other owner nearly induces his manager to maximize sales αj = ε with ε 0. As a consequence, on the second stage manager j can be labeled as an aggressive player and manager i as a passive player. To see this, we have to look at the managers objective functions EO i µ i =S L + S p i α i µ i. 10 and B i are chosen so that expected compensation just equals manager i s reservation value. 8 For p A i see, e.g., Hirshleifer and Riley 1992, pp ; Baye et al. 1996; Konrad et al For p B i logit form see, e.g., Dixit 1987, p. 893; Hirshleifer and Riley 1992, pp ; Baik and Kim 1997; Schoonbeek The proofs of the propositions are relegated to the Appendix. 10 Note that O i = α i S i µ i +1 α i S i. Intuitively, we can interpret α i as the fee which manager i has to pay for using one unit of µ i. 4

6 Substituting i by j in EO i µ i gives the objective function of manager j. Thus, α j <α i means that manager j can spend resources up to µ j = S/α j, 11 whereas manager i s upper bound is µ i = S/α i < µ j. Therefore, manager j can choose µ j slightly above S/α i to win the contest for a certainty. In equilibrium, however, manager j randomly chooses µ j out of the interval [0, S/α i ]. Contrary to this, manager i chooses to drop out of the contest i.e., µ i =0with probability G i 0 = 1 α j/α i and to take a value out of the interval 0, S/α i ] by random with the rest of the probability mass. It is interesting to see that owner j directly influences manager i s drop-out probability G i 0. Since owner j chooses α j 0, manager i almost surely drops out of the contest i.e., G i 0 1. This also explains the expected profits of the two owners: EΠ i S L and EΠ j S L S. Thus, the owner with the aggressive manager is strictly better off than the owner with the passive manager. Proposition 2 When p i = p B i then there exists a continuum of subgame perfect equilibria where the owners choose α 1 + α 2 =2on the first stage and the managers µ 1 = S α 2 and µ α1 + α = S α 1 1 α1 + α2 2 with S = S H S L on the second stage. The results of Proposition 2 show that the logit-form contest clearly differs from the case considered in Proposition 1. Now, there exists a wide range of equilibria which also contain the symmetric equilibrium where both owners induce their managers solely to maximize profits, i.e. α 1 = α 2 =1. This leads to µ 1 = µ 2 = 1 S and the following expected profits: 4 µ i EΠ i = S L + S µ i + µ j µ i = S L + 1 S Manager j does not want to spend more than µ j, because by choosing µ j =0he can achieve S L as a kind of fall-back position. 5

7 i =1, 2. On the other hand, combining 1 and 2 we obtain for the asymmetric equilibria with α 1 α 2, µ 1 µ 2, and α 1 =2 α 2: EΠ i = S L + S α i. 3 Since α 1+α 2 =2, in an asymmetric equilibrium one owner puts a positive weight on sales and the other owner a negative weight. The owner with the positive negative weight on sales, who chooses α i < 1 α j > 1, is strictly better worse off than in the symmetric equilibrium. Equation 3 shows that the most attractive equilibrium for owner i is characterized by α i 0 which implies EΠ i S L + 1 S. Therefore, owner i will receive the individually best result if he nearly 2 induces his manager to maximize sales which makes manager j dropping out of the contest α i 0 implies µ j 0. Interestingly, this asymmetric equilibrium yields similar outcomes compared to the asymmetric equilibria of Proposition 1: The expected profits of owner i with the aggressively behaving manager i.e., α i 0 and µ i 1 S tend to S 2 L + 1 S, whereas the expected profits of 2 owner j with the passive manager i.e., α j 2 and µ j 0 tend to the loser prize of the contest, S L. 6

8 3 Conclusions Standard principal-agent considerations indicate that owners should make their managers maximize profits and not sales. The results of this paper have shown that in the context of strategic interactions between managers on markets which can be characterized as a kind of contest the opposite result may hold: It can be rational for an owner to induce his manager to maximize sales. In this way, the owner s manager behaves very aggressively so that the competing manager of the other firm tends to drop out of the contest. The model considered here assumes that the two owners choose their strategic incentives simultaneously. Dropping this assumption indicates that there is a strong first-mover advantage for an owner to make his manager be the aggressive one in the market. 7

9 Appendix Proof of Proposition 1: Consider first the subgame on the second stage. Here, manager i i =1, 2 has to decide about µ i for a given pair of α 1,α 2. Manager i s objective function can be described by EO i µ i =S L + S p i αi µ i A1 with S = S H S L, because O i = α i Π i +1 α i S i = α i S i µ i + 1 α i S i = S i α i µ i. Since p i = p A i, it is obvious that there does not exist an equilibrium in pure strategies on the second stage of the game. Let, for example, α 2 <α 1 so that manager 1 can at most spend µ 1 = S/α 1 resources in the contest without making a loss, whereas manager 2 s upper bound is µ 2 = S/α 2 > µ 1. When manager 1 chooses µ 1 [0, S/α 1 ], manager 2 s best response is to choose µ 2 = µ 1 + ε with ε 0. But then manager 1 wants to choose µ 1 = µ 1 +2ε and so on. This upbidding would stop when manager 2 reaches manager 1 s upper bound so that now manager 1 s best response is µ 1 = ε with ε 0. But then manager 2 would choose µ 2 =2εwith ε 0. Thus, there cannot exist an equilibrium in pure strategies. Analogous considerations hold for α 2 >α 1 and α 2 = α 1. To derive an equilibrium in mixed strategies let G i µj i, j =1, 2; i = j be the probability that manager i spends µ j or less resources. Then, A1 can be written as EO i µ i =S L + S G j µ i α i µ i. A2 Consider the case above, where α 2 <α 1 so that manager 2 is able to win the contest in a certainty by spending resources marginal larger than S/α 1 which would imply EO 2 µ 2 = S L + S α 2 S α 1. In this case, manager 1 would 8

10 choose µ 1 =0which leads to EO 1 µ 1 =S L. Thus, an equilibrium in mixed strategies must meet two conditions: S S L + S G 1 µ 2 α 2 µ 2 = S L + S α 2 A3 α 1 G 1 µ 2 = 1 α 2 + α 2µ 2 α 1 S S L + S G 2 µ 1 α 1 µ 1 = S L G 2 µ 1 = α 1µ 1 S. A4 For α 2 >α 1, we only have to interchange the indices in A3 and A4. Altogether, asymmetric equilibria on the second stage can be characterized by G i µ =1 α j α i + α j S µ and G j µ = α i S µ A5 with α j <α i and µ [0, S/α i ]i, j =1, 2; i j. For the symmetric case α 2 = α 1 the equilibrium mixed strategies has to meet EO i µ i =0=EO i µ j = S α j S L S L + S G j µ i α i µ i i, j =1, 2; i j which leads to = S α i = G j µ = α i S µ A6 with µ [0, S/α] and α = α i = α j i, j =1, 2; i j. On the first stage, the two owners have to decide about α 1 and α 2 anticipating the possible asymmetric see A5 or symmetric see A6 outcomes for the second stage. The owners want to maximize expected profits EΠ 1 α 1 and EΠ 2 α 2, where EΠ i α i =S L + E µi [ S p i µi ] i =1, 2. In the case of an asymmetric equilibrium on stage 2 we obtain EΠ i α i = S L + S/α i [ S G j µ i µ i ] G i µ i dµ i A7 = S L + 0 S/α i 0 [ S α ] i S µ αj i µ i S dµ i = S L + α j α i 1 S 2α 2 i 9

11 and EΠ j α j = S L + S/α i [ S Gi µj µj ] G j µj dµj A8 0 = S L + 2α i α j 1 S 2α i with α j <α i i, j =1, 2; i j. The case of a symmetric equilibrium yields EΠ i α i =S L + S/α i 0 [ S α ] i S µ αj i µ i S dµ i = S L + α j α i 1 S 2α 2 i A9 with α i = α j. First, consider the asymmetric case with α j <α i. Here, owner i will choose α i =2to maximize EΠ i α i according to A7, whereas owner j will choose α j = ε with ε 0 see A8. Approximately, this would result in EΠ i α i =2,α j =0 A7 = S L and EΠ j α i =2,α j =0 A8 = S L S. A10 In the symmetric case, α i =2will maximize A9. Therefore, the owners will choose α i = α j =2and receive EΠ i α i = α j =2=EΠ j α i = α j =2 A9 = S L + 1 S. A11 4 Obviously, the two asymmetric cases α i = 2 and α j = 0 i, j =1, 2; i j correspond to two subgame perfect equilibria, whereas the symmetric case α i = α j =2does not. When owner i chooses α i =2, owner j has three alternatives: 1. He can choose α j < 2 and receives S L S with α j Or he can choose α j =2and gets S L S according to A Owner j can choose α j > 2 so that the owners reverse their roles and j approximately obtains S L S according to A7 with interchanged indices with α j =2+ε and ε 0. On the other hand, when owner j chooses an arbitrarily small but positive number 10

12 α j = ε with ε 0, owner i must choose one of three alternatives: 1. He can choose α i >εand receives S L for ε 0 according to A7. 2. Owner i can choose α i = ε, but then his expected profits tend to according to A9 with α i = α j = ε and ε Owner i can choose α i <εso that according to A8 with interchanged indices his expected profits tend to due to ε 0. The symmetric case cannot be stable, because each owner has an incentive to deviate from α i = α j =2. Proof of Proposition 2: In the logit-form case, manager i s objective function on the second stage can be written as µ i EO i µ i =S L + S α i µ µ i + µ i A12 j i, j =1, 2; i = j. The first-order conditions EO 1 µ 1 =0and EO 2 µ 2 =0 yield 12 µ 1 + µ 2 = S µ2 S µ1 =, A13 α 1 α 2 which implies α 1 µ 1 = α 2 µ 2 µ 2 = α 1µ 1 α 2. Substituting for µ 2 in A13 gives 13 µ 1 = S α 2 α 1 + α 2 2 and µ 2 = S α 1 α 1 + α 2 2. A14 On the first stage, owner i maximizes µ i EΠ i α i =S L + S µ i + µ j µ i A15 where µ i and µ j are described by A14. Substituting for µ i and µ j in A15, the first-order conditions EΠ 1 α 1 =0and EΠ 2 α 2 =0for the two owners show that α 1 + α 2 =2must hold in equilibrium The second-order conditions are met. 13 The other solution for µ 1 and µ 2 is not feasible. 14 The second-order conditions hold for α 1 + α 2 =2. 11

13 References Baik, K.H. and I.-G. Kim, 1997, Delegation in contests, European Journal of Political Economy, 13, Baye, M.R., Kovenock, D. and C.G. de Vries, 1996, The all-pay auction with complete information, Economic Theory, 8, Dixit, A., 1987, Strategic behavior in contests, American Economic Review, 77, Fershtman, C. and K.L. Judd, 1987, Equilibrium incentives in oligopoly, American Economic Review, 77, Hirshleifer, J. and J.G. Riley, 1992, The analytics of uncertainty and information, Cambridge Cambridge University Press. Konrad, K.A., Peters, W. and K. Wärneryd, 2000, Delegation in first-price allpay auctions, SSE/EFI Working Paper Series in Economics and Finance, no. 316, Stockholm School of Economics. Schoonbeek, L., 2000, A delegated agent in a winner-take-all contest, University of Groningen, mimeo. Sklivas, S.D., 1987, The strategic choice of managerial incentives, Rand Journal of Economics, 18,