On Surplus-Sharing Rules in Partnerships

Transcription

1 On Surplus-Sharing Rules in Partnerships Özgür Kıbrıs Arzu Kıbrıs SabancıUniversity March 3, 204??? Corresponding author: Faculty of Arts and Social Sciences, Sabanci University, 34956, Istanbul, Turkey. Tel: Fax:

2 Abstract This paper offers a perspective on partnership contracts in environments where the partners effort levels are verifiable and the partnership can probabilisticaly succeed or fail. We inquire how partnership contracts can be used to incentivize the partners to maximize effort i.e. to achieve productive effi ciency and whether maximizing effort really maximizes the partners welfare. We consider a family of partnership contracts which are based on two surplus-sharing schemes commonly used in real life, namely proportional sharing and equal sharing. The contracts we consider also allow using different surplus-sharing schemes in cases of positive and negative surplus. We show that the productively effi cient contract allocates a positive surplus proportional to efforts while it allocates a negative surplus equally among the partners. This contract does not maximize egalitarian social welfare, however. There is a continuum of egalitarian optimal partnership contracts, each of which using the same mixture of proportional and equal surplus-shares in cases of positive and negative surplus and different egalitarian optimal contracts differing in the weights that they use. Such contracts all induce the same utility for all partners and they all create the same total effort. But they differ in terms of individual partners effort choices. Keywords: partnerships, partnership contracts, surplus-sharing, noncooperative effort choice, proportional surplus shares, equal surplus shares, productive effi ciency, egalitarian social welfare, utilitarian social welfare. JEL classification codes: C72, D86, L22, J33 2

3 Introduction A look at real life partnership contracts reveals two interesting features regarding surplussharing. First, it is not uncommon to see equal sharing of the surplus, even in partnerships where good measures of each partner s contribution are available. Second, the same surplussharing scheme is used in cases of both positive and negative surplus. In fact, if only the surplus-sharing scheme to be used in case of positive surplus is specified, the legal default is that the same surplus-sharing scheme is used in case of negative surplus as well and vice versa. These two features are so common that in many countries, they are part of the state partnership act, that is, the statutory partnership agreement that applies to any partnership that does not have a written agreement e.g. see McMasters, 203. In this paper, we investigate the value of these two features and ask whether the partners can improve over them. In most professional service partnerships such as in law, medicine, accounting, or real estate, there are good measures of the partners effort choices. For example in law partnerships, information on the number of hours billed, number of cases won, or number of new customers brought in serve either individually or in an aggregated way as a measure of effort a lawyer puts in. Similarly, in medical partnerships, data on the number of patients treated, number of hours billed, or the number of surgeries performed are available measures of effort. In professional service partnerships, the partners efforts combined with their human capital serve as the main input for the success of the partnership and thus, can not be considered negligible information in profit sharing. Yet, fixed surplus-shares are observed quite frequently in professional service partnerships. Farrell and Scotchmer 988 and Lang and Gordon 995 mention three basic surplussharing schemes by which profit is divided among law partners. First, all members of the same seniority receive the same fixed profit share. This equal sharing of the surplus also called the lock-step system is used by most 2 or 3 partner law firms, which account for two thirds of all law firms, although less than half the lawyers also see Curran, 985; Flood, 985. Second, firm s founders decide surplus-sharing based on a subjective evaluation of performance. Third, the partners use some objective formula that compensates according to Finally, in real estate partnerships, information on the number of houses sold, number of hours spent meeting with customers, or the number of new houses brought in can be used to create a measure of the effort each partner puts in the partnership. 3

4 time billed, business brought in, public stature, etc.. These second and third type of schemes are sensitive to effort, very much like proportional sharing. Empirical studies show that fixed surplus-sharing schemes can be detrimental to the partnership. For example, Newhouse 973 finds that shirking exists in medical group practices that use equal-sharing arrangements and that shirking increases in partnership size. Reinhardt, Pauly, and Held 979 find that productivity-based surplus-sharing arrangements increase productivity in medical partnerships. Leibowitz and Tollison 980 find that there is shirking under equal-sharing in legal practices. Finally, the case of Andersen Worldwide provides a well-known example. Founded in 94 as an accounting firm, it was divided into two in 998, as the consulting and audit-tax arms of the firm disputed with each other for many years over the division of profits. Huddart and Liang 2003 note that the profit sharing formula in Andersen was designed to level partner compensation across units of the firm and this invited shirking, increasing conflict, and finally leading to the dissolution of the two arms. Motivated by the above observations, this paper offers a perspective on surplus-sharing schemes, hereafter, partnership contracts. We focus on environments where the partners effort levels are observable and verifiable, as exemplified above. We also assume that the partnership either makes a positive or a negative surplus, depending on a random state. 2 A partnership contract can thus condition the surplus-sharing of a partner on both i the partners effort choices, and ii whether the surplus is positive or negative. We consider a family of partnership contracts that are based on the two surplus-sharing schemes commonly observed in real life partnerships, that is, proportional and equal surplus-sharing. A typical member of this family of contracts allocates some fraction of the surplus equally among the partners, and the rest proportional to the partners effort choices. Such surplus-sharing schemes are not uncommon in real-life partnerships e.g. see Gaynor and Pauly, It is natural that the partners efforts do not perfectly determine the surplus generated. The latter might depend on external variables such as the performance of competitors, the state of the economy, or some unmodeled variables such as the number of cases won or patients saved, which in turn effects the reputation of the partnership as well as its current and future income. We consider a very simple scenario where the partnership is either succesful in which case the production function brings a net return on efforts or it fails and the production function brings in negative returns on efforts. Several previous studies use a similar production function e.g. Morrison and Wilhelm 2004, Comino, Nicolo, Tedeschi 200, Li and Wolfstetter

5 We analyze the implications of such partnership contracts on the partners effort choices as well as their welfare. Specifically, we inquire how partnership contracts can be used to incentivize the partners to maximize effort such contracts are called productively effi cient in the literature and whether maximizing effort really maximizes the partners welfare. We also analyze how a surplus-sharing scheme used in case of positive negative surplus affects the optimal choice of the other surplus-sharing scheme and how these choices are affected by important parameters such as the number and risk attitudes of partners, the uncertainty they face, returns in case of success and losses in case of failure. To this end, we analyze a simple model in which a group of partners have agreed on a partnership contract. Given the contract and full information about the others characteristics, the partners simultaneously choose their effort levels. The nature then determines whether the partnership makes a positive or a negative surplus. Depending on the type of surplus realized, the relevant part of the contract is used to compensate the partners. Each partner, when choosing his effort, tries to maximize his expected utility from the partnership. Almost all the theoretical literature on partnerships seems to have used additively separable utility functions quasilinear preferences. Similar to those studies, we measure effort in monetary units. But we alternatively assume that the agents have constant absolute risk aversion CARA utilities. Since we consider a stochastic production function, the CARA family provides us a good way to measure the affect of the agents risk attitudes on the outcome. Analysis of a simple quasilinear model confirms our findings but turns out to be less interesting in terms of the interaction among the agents. The paper is organized as follows. In the next section, we discuss the related literature. In Section 2, we present the model. In Section 3, we calculate and analyze the Nash equilibrium induced by each rule. In Section 4, we compare partnership contracts in terms of the total effort they induce in equilibrium. In Section 5, we then compare them in terms of social welfare. We summarize our findings and conclude in Section 6. The proofs are relegated to Section 7.. Literature Following the seminal works of Alchian and Demsetz 972 and Holmström 982, a big literature focuses on the question of how to design an effi cient partnership contract in environments where the partners effort choices are not verifiable, and thus, effort sensitive 5

6 schemes, such as proportionality, are not available. In contrast to Alchian and Demsetz 972 who argue that effi ciency can only be restored by bringing in a principle who monitors the agents, Holmström shows that group incentives alone can remove the free-rider problem. 3 The following literature focuses on the same question under alternative assumptions. Kandel and Lazear 992 analyze the effect of peer pressure, Legros and Mathews 993 analyze the effect of limited liability, Miller 997 and Strausz 999 analyze cases where a partner can observe the effort exerted by a subset of other partners, and Morrison and Wilhelm 2004 discuss moral hazard problems associated with intergenerational transfer of human capital. Hart and Holmström 200 and Hart 20 adopt the contracts as reference points approach to discuss shading and effi cient partnership contracts. Farrell and Scotchmer 988 analyze the effi ciency costs of equal-sharing in a theoretical model of partnership formation. There is also a big empirical literature on the effect of compensation schemes on effort and productive effi ciency in partnerships. For example, Newhouse 973, Reinhardt, Pauly, and Held 979, Leibowitz and Tollison 980 are discussed in the previous section. It is also useful to mention Gaynor and Pauly 990 who develop and test a model of the effect of alternative contracts on productive effi ciency in medical group practices. The partnership contracts analyzed are similar to those that we consider. Huddart and Liang 2003 also mention moral hazard as an important feature of partnerships. They analyze the trade-off between risk sharing and free riding in choosing the size of the partnership. Similar to us, in their model output is stochastic though the shocks are individual and the agents have CARA utilities. There are many earlier papers that, like us, model the output as stochastic. For example, see Huddart and Liang 2003, Comino, Nicolo, Tedeschi 200. Again similar to us, several earlier studies argue that the partners expectations on their shares in case the partnership fails will have an effect on the partners effort choices. For example, see Comino, Nicolo, Tedeschi 200 or Li and Wolfstetter While we work under different informational assumptions, Homström s question is very similar to this paper. Quoting pg 326: The question is whether there is a way of fully allocating the joint outcome so that the resulting noncooperative game among the agents has a Pareto optimal Nash equilibrium. Holmström shows that the free rider problem can be solved as follows. One sets an output objective by utilizing the observable information about the agents costs of effort. If it is not met, all partners receive zero. If it is met, all agents receive a bonus. 6

7 We consider partnership contracts that are based on two principles: proportional and equal sharing of surplus. These principles have played a central role in the axiomatic literature on surplus-sharing. See O Neill 982, Aumann and Maschler 985 and the following literature reviewed in Thomson 2003 and 2008 for axiomatic studies of negative surplus referred to as claims or bankruptcy problems by this literature. On the other hand, Moulin 987 and the following literature provides an axiomatic study for positive surplus. There also is a smaller literature that covers both cases simultaneously. For example, Chun 988 proposes characterizations of classes of rules that mix the proportionality and equal awards principles in both cases of positive and negative surplus. Herrero, Maschler, Villar 999 propose and analyze a rights-egalitarian solution which uses the equal awards principle in case of positive surplus and the equal losses principle in case of negative surplus. Finally it is useful to mention Kıbrıs and Kıbrıs 203, where we use a similar modeling approach to analyze the investment implications of bankruptcy laws. In that paper, the firm either makes profits and the investors receive interest in their investment or the firm goes bankrupt and the investors compensation is determined by the existing bankruptcy laws. Aside from the fact that the two papers consider two separate economic institutions, it is useful to note, from a technical point of view, that the partnership contracts considered in this paper allow for a much larger class of allocation rules in case of profits i.e. positive surplus. On the other hand, the bankruptcy rules considered in Kıbrıs and Kıbrıs 203 allows for a larger class of allocation rules than the surplus-sharing rules that we consider since not all rules in that paper are relevant for surplus allocation. The literature mentioned above is silent when it comes to analyzing the incentive effects of surplus-sharing schemes that are used in cases of positive and negative surplus. The literature also does not consider the possibility of using different surplus-sharing arrangements in the cases of positive and negative surplus. Our paper contributes to the literature on that front and. As will be detailed below, we show that the interaction between the cases of positive and negative surplus produces a number of surprising hypothesis regarding the design of partnership contracts. 7

8 2 Model Let N = {,..., n} be the set of partners. Each i N has the following Constant Absolute Risk Aversion CARA utility function u i : R + R on money: u i x = e aix. Assume that each partner i is risk averse, that is, a i > 0. Also assume that a... a n. Each partner i chooses the effort level s i R + he will contribute to the partnership. We measure effort in monetary units or equivalently assume constant marginal cost of effort and normalize it to one. The total effort expended by the partners is then N s j. With success probability p 0,, this value brings a return r 0, ] and becomes + r N s j, creating a positive surplus for the partners. With the remaining p probability, the partnership makes a negative surplus and its value becomes β N s i where β 0, is the fraction that survives failure. Remark In many real life examples of partnerships, output is not a deterministic function of effort. Consider medicine and law. In both, the success of the effort put in i.e. success in treatment of patients or legal proceedings depends on many factors outside the control of the agent. As a result, the success of partnership is not deterministic. Of course, it might be more realistic to model the success probability as an increasing function of effort. We take it to be constant to simplify our model and we treat it as a benchmark case. Remark 2 Also related to the randomness of output is the following. If the two agents face independent shocks that determine the success of their endeavours, using surplus-sharing arrangements that equate gains or losses have an additional rationale since they provide insurance for the agents. We might want to analyze this argument in the second part of the paper where we make the shocks of the two agents independent. A partnership contract is a pair of rules F, G to be used in case of positive and negative surplus, respectively. The positive-surplus rule F allocates the gross returns + r s j according to the vector of efforts s, partner i s share being F i s, + r s j. The negative-surplus rule G, on the other hand, allocates the amount that survives failure β s j according to the vector of efforts s, partner i s share being denoted as G i s, β s j. We will be interested in partnership contracts that are based on two central surplussharing rules. Suppose the partnership creates value V. From previous discussion, we know 8

9 V is either + r s j or β s j. But the next two definitions will be independent of what V is. The proportional surplus-sharing rule, P, allocates the surplus proportional to the partners effort choices. The share of a typical agent is then P i s, V = s i + s i sj V = s i sj V s j where V s j is the surplus. The equal surplus-sharing rule, E, allocates the surplus equally. The share of an agent is then E i s, V = s i + V s j n. Using convex combinations of these two, one can construct a continuum of surplus sharing rules as follows. For each ρ [0, ], the PE[ρ] rule gives weight ρ to the proportional allocation and the remaining weight to equal sharing of the surplus: P E[ρ] i s, V = ρp i s, V + ρ E i s, V = ρ s i V + ρ s i + V s j sj n = s i + V s j ρ s i + ρ. sj n As can be seen in the second expression, in the P E [ρ] rules, each partner is first reimbursed for his effort. Then the additional returns are allocated among the partners by using a mixture rule that gives ρ weight to the proportional allocation of surplus and ρ weight on equal surplus-sharing. An interpretation of this rule also mentioned in Gaynor and Pauly, 990 is that the partnership allocates ρ part of the surplus equally among the partners and uses the remaining amount ρ to give bonuses as a proportion of efforts. As noted in the introduction, a novelty of this paper is that we allow the partners to use different mixtures of proportionality and equality in cases of positive and negative surplus. The class of partnership contracts that we analyze, therefore combine a positive-surplus rule P E [γ] and a negative surplus rule P E [α] where α, γ [0, ] and α γ is allowed. We will refer to a partnership contract as P E [γ, α]. For each partnership contract P E [γ, α], we analyze the following game it induces over the partners. Given the partnership contract P E [γ, α], the partners simultaneously choose their effort levels. Agent i s expected payoff from a strategy profile s R n + is U P E[γ,α] i s = pu i F i s, + r s j s i + pu i G i s, β s j s i where F i s, + r s j s i and G i s, β s j s i are his surplus shares in cases of positive and negative surplus, respectively. Let U P E[γ,α] = U P E[γ,α],..., Un P E[γ,α]. The partnership game induced by contract P E [γ, α] is then defined as G P E[γ,α] = R N +, U P E[γ,α]. 9

10 Let ɛg P E[γ,α] denote the set of Nash equilibria of G P E[γ,α]. To measure the partners social welfare from a contract, we will resort to two leading alternative social welfare measures in the literature. The egalitarian social welfare level induced by a partnership contract P E [γ, α] is the minimum utility an agent obtains at the Nash equilibrium of the partnership game induced by P E [γ, α]: EG P E[γ,α] p, r, β, a,..., a n = min i N U iɛ G P E[γ,α]. The utilitarian social welfare level induced by a partnership contract P E [γ, α] is the total utility the agents obtains at the Nash equilibrium of the partnership game induced by P E [γ, α]: UT P E[γ,α] p, r, β, a,..., a n = i N U i ɛ G P E[γ,α]. 3 Equilibrium As defined in the previous section, each partnership contract induces a partnership game among the agents. We start by analyzing the Nash equilibria of each such game. This section serves as a preliminary for our comparisons of total effort in Section 4 and social welfare in Section 5. The following proposition determines the form of the unique Nash equilibrium under P E [γ, α]. It assumes Inequality which requires that the partners are not too different in terms of their risk attitudes. 4 Its left hand side has played an important role in previous studies such as Wilson 968 and Huddart and Liang 2003, and it is interpreted as a partner s risk tolerance relative to the risk tolerance of the partnership e.g. see Wilson s interpretation for the case of syndicates. Inequality excludes parameter values for which at the Nash equilibrium an agent s compensation in case of negative surplus is negative, that is, it identifies the parameter values under which P E[α] i s 0 for each i N. As a result, Inequality also excludes parameter values for which at the Nash equilibrium some partners choose positive effort levels while others choose zero. 4 Since agent n is the most risk averse partner, Inequality guarantees this inequality for every partner i. 0

11 Proposition Nash equilibrium under P E [γ, α] Assume a n N r α β rα + αβ + rαβ rβγ +. αβ α + n r β + If ln prnγ γ+ β pnα α+ equilibrium 0,..., 0. If ln prnγ γ+ β pnα α+ is a Nash equilibrium. 0, the partnership game under P E [γ, α] has a unique Nash > 0, α = 0, and γ = 0, any effort profile s such that pr n ln s p β i = a n β + r N Otherwise, the game has a unique Nash equilibrium s in which each agent i chooses a positive effort s i s i = given by n r β + a i r α β rγ + αβ + N n r β + α + rγ αβ and which satisfies P E[α] i s 0 for each i N. ln pr nγ γ + β p nα α + 2 Since a... a n, Equation 2 implies s... s n in case of a unique Nash equilibrium. We will therefore say that agent i is a bigger partner than agent j whenever i j. Regarding the ln term in the proposition, note that if pr nγ γ + > β p nα α +, all partners choose positive efforts at the Nash equilibrium. This condition simply compares the return on unit effort in case of positive surplus, r nγ γ +, weighted by the probability of success, p, with the loss incurred on unit effort in case of negative surplus, β nα α +, weighted by the probability of failure, p. Exerting positive effort is optimal if the returns in case of success outweigh the losses incurred in case of failure. 5 5 To see this, divide both sides by n and obtain p γr + γ r n > p α β + α β n..

12 The following result identifies conditions under which the partnership game has a dominant strategy equilibrium. Partnership contracts that induce dominant strategy equilibria are advantageous to those that do not since it is possible to make a stronger prediction about how the partners will behave. Proposition 2 Dominant strategy equilibrium under P E [γ, α] The partnership game induced by the contract P E [, ] has a dominant strategy equilibrium in strictly dominant strategies. No other partnership contract induces dominant strategy equilibria. This result is a corollary of Proposition. It follows from Equation 2 that the partnership games induced by P E [γ, α] contracts admit dominant strategy equilibria if and only if r α β rγ + αβ + = 0 in which case partner i s best response is independent of the other agents effort choices. This equality holds if and only if α = γ =. Note that ln > 0, α = 0, and γ = 0 constitutes a limit case where prnγ γ+ β pnα α+ there is multiple equilibria. However, this multiplicity will not be an issue for the rest of our analysis since all such equilibria produce the same total effort level. 3. A Numerical Example Next, we present a numerical example to demonstrate how individual effort choices depend on the partnership contract P E [γ, α]. In the example, the parameter values are r = 0.3, p = 0.8, β = 0.7, a =, a 2 =.5, γ = 0.5. Figures 3. and 3. plot how individual efforts change as a function of α, the weight the negative-surplus rule assigns to proportionality. As can be seen in Figure 3., an increase in α decreases Partner s effort. This might seem surprising at first glance, since it is commonly argued in the literature that a shift from equal surplus shares to proportional surplus share will increase individual effort. However, the reader will note after a closer inspection that The left hand side expression γr + γ r n has two parts. The γ weighted part r is the partner s return under proportional surplus-sharing and the γ weighted part r n is his return under equal surplus-sharing. Thus, the weighted average γr + γ n r is the partner s return under the positive-surplus rule P E [γ]. The right hand side expression α β + α β n again has two parts. The α weighted part of this expression, β is the loss incurred for unit effort in case of proportional surplus-sharing and the α weighted part β n is the loss incurred in case of equal surplus-sharing. Thus, the weighted average α β + α β n is the partner s losses under the negative-surplus rule P E [α]. 2

13 Partner, being the bigger partner, receives a lower surplus share as the negative-surplus rule moves equal to proportional shares. This, in turn, decreases his effort choice. s_ Partner s equilibrium effort choices, as a function of α. Maybe more surprisingly, Figure 3. shows that α has a non-monotonic effect on the effort choice of the smaller partner, Partner 2. In the following picture, as α moves the negative-surplus rule from equal surplus-sharing towards proportional sharing, Partner 2 first increases and then decreases his effort choice. It is useful to note that the shape of this graph is sensitive to γ, the weight of proportionality in the positive-surplus rule. For example, for small γ values, this graph would be increasing and for high γ values, it would be decreasing. This shows that the incentives Partner 2 face are not straightforward, but are determined through and interplay of the positive-surplus and negative-surplus rules. alpha s_ alpha Partner 2 s equilibrium effort choices, as a function of α. In the above example, we next fix α = 0.3 and let γ vary. Figure 3. demonstrates that 3

14 a similar issue arises with respect to the effect of changes in γ. Equilibrium effort choices of the two partners then look like as follows. s_ gamma Partner s equilibrium effort choices, as a function of γ. s_ gamma 0.4 Partner 2 s equilibrium effort choices, as a function of γ. As can be seen above, a move in the success rule from equal surplus-sharing to proportionality increases Partner 2 s effort choice, as claimed by the previous literature. However, the effect on the bigger partner, Partner is non-monotonic. Thus, contrary to what the previous literature suggests, moving from a fixed surplus-sharing rule towards proportional shares the piece-rate does not necessarily increase individual effort for all partners. Once again, the effect depends on the interplay between the success and failure rules. For example, if we fixed a higher value of α to start with, we could obtain an example where Partner s effort choice is also increasing in γ. 4

15 s_ + s_ gamma Figure : The effect of γ on total effort. The parameters are r = 0.3, p = 0.8, β = 0.7, a =, a 2 =.5, α = Productive Effi ciency In this section, we compare partnership contracts in terms of the total effort that they induce in equilibrium. Our objective is to identify the productively effi cient partnership contracts, that is, contracts that maximize output by maximizing total effort that the partners choose. In the above numerical example, the choice of γ, that is, the weight of proportionality in the positive-surplus rule, affects total effort as in Figure. The figure confirms the common belief that a move from equal surplus-sharing towards proportionality increases total effort. For the same parameter values, Figure 2, shows the effect of α, that is, the weight of proportionality in the negative-surplus rule, on total effort. Surprisingly, an increase in α decreases total effort. As demonstrated in the previous section, a look at individual effort choices suggests no clear prediction as to how the total effort level would be affected from changes in the underlying partnership contract. On the other hand, the above figures suggest that, in terms of total effort, partnership contracts have a clear ordering. The following theorem shows that what we observe in this numerical example in fact generalizes to the whole parameter space. Theorem Assume. Equilibrium total effort under the P E [γ, α] family of contracts 5

16 s_ + s_ alpha Figure 2: The effect of α on total effort. The parameters are r = 0.3, p = 0.8, β = 0.7, a =, a 2 =.5, γ = 0.5. is i nondecreasing in γ the weight of proportionality in the positive-surplus rule and ii nonincreasing in α the weight of proportionality in the negative-surplus rule. Both effects are strict if ln > 0. Furthermore, both effects are increasing in prnγ γ+ β pnα α+ the number of partners in the partnership. In terms of what it says regarding the positive-surplus rule, the theorem supports the general view that moving from a fixed surplus-sharing rule like E to proportional division increases the total effort the partners put into the partnership. For the negative-surplus rule, however, the theorem surprisingly identifies that a move away from proportional division towards equal surplus-sharing increases the total effort the partners put into the partnership. The theorem, thus, shows us that a way to improve over the commonly-used piece rate contract is to change the surplus-sharing rule used in case of negative-surplus; a move towards equal surplus shares helps productive effi ciency. It is also important to note that the effect of the contract on total effort is emphasized in partnerships with a greater number of partners. As a corollary of the above theorem, we obtain a strong result regarding productive effi ciency. Specifically, we observe that among all the partnership contracts that we consider, there is a unique productively effi cient contract. 6 It is the P E [, 0] contract which uses

17 proportional division in case of positive-surplus and equal surplus-sharing in case of negativesurplus. Corollary 3 Assume. Then P E [, 0] is a productively effi cient contract in the P E [γ, α] family. Furthermore, if ln > 0, P E [, 0] is the unique productively effi cient contract. prnγ γ+ β pnα α+ It is interesting to note that, even when all partners are identical in terms of risk aversion, the ordering of partnership contracts in terms of total effort is still as above. Particularly, P E [, 0] still remains as the unique productively effi cient contract. This means that the P E [γ, α] contracts significantly differ in terms of the incentives that they provide even in symmetric partnerships i.e. when all partners are identical in terms of their risk attitude. 5 Individual and Social Welfare In this section, we look at the individual and social welfare levels induced by alternative partnership contracts. We are able to make an analytical comparison in terms of egalitarian social welfare. Additionally, we will carry out a numerical analysis in terms of utilitarian social welfare. The following pictures demonstrate how equilibrium welfare of two partners change as the weights α and γ change in their P E [γ, α] partnership contract. The parameter values are r = 0.3, p = 0.8, β = 0.7, a =, a 2 = 3, γ = 0.5. The first picture fixes γ = 0.5 and the second one fixes α = 0.5. u_ and u_2 0.7 alpha

18 Utility of Partner black and Partner 2 red as a function of α. The weight of the success rule γ = 0.5. u_ and u_2 0.7 gamma Utility of Partner black and Partner 2 red as a function of γ. The weight of the failure rule α = 0.5. The following observations are in order. First, in both pictures Partner the bigger partner receives a smaller utility than Partner 2 if and only if α < γ, that is, when the success rule is closer to proportionality than the failure rule. This means that in this example, egalitarian social welfare is equal to the utility of Partner when α < γ and to the utility of Partner 2 when α > γ. As both pictures demonstrate, when α = γ, the two partners receive equal payoff. Second, this egalitarian social welfare increases as α and γ gets closer to each other, and is maximized at α = γ. Surprisingly, both of the above points are generalizable to an arbitrary number of agents and to all parameter values we consider. The following proposition orders the agents according to their equilibrium welfare. Proposition 4 Assume. Under the P E [γ, α] family of partnership contracts, the partners are ordered according to their equilibrium utilities as, 2,..., n. If α > γ, the least riskaverse Partner always receives the smallest utility and the most risk-averse Partner n always receives the highest utility. If α < γ, the ordering is reversed, Partner now receiving the highest utility and Partner n, the smallest. Finally, if α = γ, all agents receive the same utility level. The above proposition implies that the egalitarian social welfare is equal to the equilibrium payoff of either the most or the least risk averse partner, depending on the α-γ 8

19 relationship in their partnership contract. The following theorem shows that this egalitarian social welfare is maximized at α = γ. Theorem 2 Assume. Under the P E [γ, α] family of partnership contracts, egalitarian social welfare is decreasing in α γ. When α = γ = x > 0, all partners payoffs are equal and this common payoff, which is also the egalitarian social welfare under the P E [x, x] partnership contract, is independent of x. While all P E [x] P E [x] partnership contracts induce the same egalitarian social welfare level, they might be different in other aspects. The first that comes to mind is the agents effort choices. This does not turn out to be the case though, as remarked below. Remark 3 All P E [x, x] partnership contracts induce same total effort level in equilibrium. They are thus identical with respect to productive effi ciency. For a proof, please see the Appendix. These contracts, however, differ in terms of the individual effort choices that they induce in equilibrium. Partners who are less more risk averse than the average decrease increase their effort choices in response to an increase in the common x, keeping total effort constant. We conclude this section with a discussion of utilitarian social welfare. For this case, the ordering of partnership contracts in terms of utilitarian social welfare depends on the underlying parameter values. This makes a general analytical result as in the case of egalitarian social welfare infeasible. However, a numerical analysis shows that utilitarian social welfare ordering of contracts is not radically different than egalitarian social welfare ordering. We first carried out a numerical analysis for the case of two partners. We allowed the following parameter values: β, p, r, α, γ {0.0, 0., 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99}, a, a 2 {0., 0.5,,.5, 2, 2.5, 3, 3.5, 4, 4.5, 5} and a 2 a. After eliminating parameter combinations that violate Inequality as well as those that produce zero effort in equilibrium, we end up with parameter combinations. Surprisingly, at that is, at 97.7 % of these parameter combinations, utilitarian social 9

20 welfare is maximized when γ = α. At the remaining parameter combinations, utilitarian social welfare is maximized at a γ α and the two cases γ > α and γ < α are observed at almost equal frequency. At that is, 22.5 % of this , the difference between γ and α is. We also carried out a numerical analysis for the case of three partners. Since the computer could not handle the above grid, we switched to a slightly coarser grid of β, p, r {0.0, 0.6, 0.3, 0.46, 0.5, 0.66, 0.7, 0.86, 0.9}, α, γ {0.0, 0., 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99}, a {0., 0.7,.3,.9, 2.5, 3., 3.7, 4.3, 4.9} and a 3 a 2 a. After eliminating parameter combinations that violate Inequality as well as those that produce zero effort in equilibrium, we end up with parameter combinations. Similar to the two-partner case, at that is, at 98.9 % of these parameter combinations, utilitarian social welfare is maximized when γ = α. At the remaining parameter combinations, utilitarian social welfare is maximized at a γ α; at of which γ > α and at 505 of which, γ < α. At that is, 28.4 % of this 8 459, the difference between γ and α is. These numerical findings should be interpreted with caution. The γ = α finding, in a significant number of the cases, is due to the grid that we impose on the parameter space. Thus, we can only deduce from this analysis that in most cases, utilitarian social welfare is maximized at α, γ values that are close to each other and that, maximizing utilitarian social welfare does not create contracts that are radically different than those that maximize egalitarian social welfare. 6 Conclusion Our analysis compares a family of partnership contracts i.e. surplus allocation rules in terms of total effort and social welfare that they induce in equilibrium of a noncooperative effort-choice game. Our findings are as follows: i Independent of the parameter values considered, equilibrium total effort induced by a partnership contract is increases as the positive-surplus rule gets closer to proportionality 20

21 and the negative-surplus rule gets closer to equal surplus-shares. Using proportionality in case of positive surplus and equal-surplus shares in negative surplus maximizes total effort. ii Independent of the parameter values considered, egalitarian social welfare increases as the weights of proportionality used in cases of positive and negative surplus i.e. γ and α get closer to each other. Partnership contracts where γ = α all maximize egalitarian social welfare. Such contracts give all agents the same welfare and produce the same amount of total effort. They, however, are different in terms of individual effort choices that they induce. iii The ordering of partnership contracts in terms of utilitarian social welfare depends on the parameter values. Thus a general statement as in egalitarian social welfare or total effort can not be made. However, a numerical analysis shows that the utilitarian optimal partnership contracts are not radically different than egalitarian optimal ones. Simulations for two and three agent partnerships show that at around 98 % of the parameter space, utilitarian social welfare is maximized when γ = α. iv In symmetric games where a =... = a n, the egalitarian optimal contracts described in ii additionally Pareto dominate all other contracts. v There always is a unique dominant strategy equilibrium under the partnership contract which uses proportionality when allocating both positive and negative surplus. P RO. No other partnership contract induces dominant strategies. Overall, we observe a trade-off between maximizing total effort and social welfare. To maximize total effort, two opposite surplus allocation rules needs to be used in cases of positive and negative surplus. However, maximizing egalitarian social welfare and as our numerical analysis reveals, utilitarian welfare to a certain extent requires choosing the same surplus-allocation rule in cases of both positive and negative surplus. For tractability of the model, we use CARA utility functions. While the CARA family is widely used in economic modeling as well as finance, it is an open question whether our findings are replicated with other families of utility functions. In our model, the rate of return r and the probability of success p are independent of the partners effort levels. This is intended to serve as a benchmark for the more realistic case where these parameters depend on effort. The latter question is left for future research. 2

22 References [] Alchian, A.A., Demsetz, H. 972, Production, Information Costs, and Economic Organization, American Fconomic Review, 625, [2] Aumann, R. J. and Maschler, M. 985, Game Theoretic Analysis of a Bankruptcy Problem from the Talmud, Journal of Economic Theory, 36, [3] Chun, Y. 988 The proportional solution for rights problems, Mathematical Social Sciences, 5, [4] Comino, S., Nicolo, A., Tedeschi, P. 200, Termination clauses in partnerships, European Economic Review, 54, [5] Curran, B. 985, The Lawyer Statistical Report: A Statistical Profile of the U.S. Legal Profession in the 980s, American Bar Foundation, dist. Hein and Co. Inc. [6] Farrell, J., Scotchmer, S. 988, Partnerships, Quarterly Journal of Economics, 032, [7] Flood, J. 985, The Legal Profession in the United States, The American Bar Foundation, Chicago, IL. [8] Gaynor, M., Pauly, M.V. 990, Compensation and Productive Effi ciency in Partnerships: Evidence from Medical Group Practice, Journal of Political Economy, 983, [9] Hart, O. 20, Thinking about the Firm: A Review of Daniel Spulber s The Theory of the Firm., Journal of Economic Literature, [0] Hart, O. and Homström, B. 200, A Theory of Firm Scope, Quarterly Journal of Economics, 252, [] Herrero, C., Maschler, M., & Villar, A. 999, Individual rights and collective responsibility: the rights egalitarian solution, Mathematical Social Sciences, 37, [2] Holmström, B. 982, Moral Hazard in Teams, Bell Journal of Economics, 3,

23 [3] Huddart, S. and Liang, P.J. 2003, Accounting in Partnerships, American Economic Review, 93:2, [4] Kandel, E. and Lazear, E. 992, Peer Pressure in Partnerships Journal of Political Economy, 004, [5] Kıbrıs, Ö., & Kıbrıs, A On the investment implications of bankruptcy laws, Games and Economic Behavior, 80, [6] Lang, K., Gordon, P.J. 995, Partnership as insurance devices: theory and evidence, RAND Journal of Economics, 264, [7] Legros, P., Matthews, S.A. 993, Effi cient and Nearly-Effi cient Partnerships, The Review of Economic Studies, 603, [8] Leibowitz, A., Tollison, R. 980, Free Riding, Shirking, and Team Production in Legal Partnerships, Economic Inquiry, 8, [9] Li, J. and E. Wolfstetter 200, Partnership failure, complementarity, and investment incentives, Oxford Economic Papers, 62, [20] McMasters Accountants, Solicitors, Financial Planners 203, The practice managers guide to co-ownership agreements, partnerships, and associateships, [2] Miller, N.H. 997, Effi ciency in Partnerships with Joint Monitoring, Journal of Economic Theory, 77, [22] Morrison, A., Wilhelm, W.J. 2004, Partnership Firms, Reputation, and Human Capital, American Economic Review, 945, [23] Moulin, H., 987, Equal or proportional division of a surplus, and other methods, International Journal of Game Theory, 6:3, [24] Newhouse, J.P. 973, The Economics of Group Practice, The Journal of Human Resources, 8,

24 [25] Reinhardt, U.E., Pauly, M.V. and Held, P.J. 979, Financial Incentives and the Economic Performance of Group Practitioners In Analysis of Economic Performance in Medical Group Practices, edited by Philip J. Held and Uwe E. Reinhardt. Project Report no Princeton, N.J. [26] O Neill, B. 982 A Problem of Rights Arbitration from the Talmud, Mathematical Social Sciences, 2, [27] Thomson, W Axiomatic and Game-Theoretic Analysis of Bankruptcy and Taxation Problems: a Survey, Mathematical Social Sciences, 45, [28] Thomson, W. 2008, How to Divide When There Isn t Enough: From Talmud to Game Theory, unpublished manuscript. [29] Wilson, R. 968, The Theory of Syndicates, Econometrica, 36, Appendix Proof. Proposition Under the family P E [γ, α], the positive-surplus rule is and the negative-surplus rule is P E [γ] i s, + r s j = s i + γrs i + γ r s j n. P E[α] i s, β s j = Thus, the utility function of partner i is U P E[γ,α] i s = pe nαβ + n + β α α β s i n n rs i +r s j a i γrs i + γ N\i n s j. N\i pe β+n α n a i s i + α β a n i N\i s j. 24

25 Its first derivative is U P E[γ,α] i s i, s i = s i n pra i nγ γ + exp a i rγs i rs i + s j r γ n N\i s j n exp N\i a i α β n n a is i α n + β a i β p nα α + and the second derivative is 2 Ui LP s i, s i = s 2 i n 2 pr2 a 2 i nγ γ + 2 exp a i rγs i rs i + s j r γ n N\i + s j n exp N\i 2 a i α β n n a is i α n + β < 0. a 2 i β 2 p nα α + 2 Equating the first derivative to zero, we obtain as a function of s i : s i = σ i s j prnγ γ+ n ln β pnα α+ = a i α + nα β + n rγ + r r α β rγ + αβ + α + nα β + n rγ + r N\i Partner i s best response is the maximum of 0 and this expression. The slope of this linear expression is always negative. Also note that the constant prnγ γ+ n ln part β pnα α+ > 0 if and only if ln prnγ γ+ a i > 0. Therefore, if α+nα β+n rγ+r β pnα α+ ln 0, then the unique Nash equilibrium is s = 0,..., 0, giving us the prnγ γ+ β pnα α+ first case. From now on, we assume ln prnγ γ+ β pnα α+ > 0. For the second case, additionally s j. 25

26 assume γ = α = 0. In this case, Condition simplifies to a n N n. prnγ γ+ n ln This implies β pnα α+ = n ln pr β p a i α+nα β+n rγ+r pr n ln β p β+r a i β+r for each i N and n ln pr for each i, j N. Also, the slope of the best response function is r α β rγ + αβ + α + nα β + n rγ + r = r β + r β + =. Thus in this case, any effort profile s such that s i = n ln establishing Case 2. s i = pr β p a n β+r β p a i = β+r is a Nash equilibrium, For the final case, additionally assume γ 0 or α 0. Then the best response functions have less than unit slope and solving them together reveals n r β + a i r α β rγ + αβ + N for each i N. n r β + α + rγ αβ nr β+ r α β rγ+αβ+ a i N ln pr nγ γ + β p nα α + We next claim 0. Note that n r β + α + rγ αβ > nr β+α+rγ αβ 0 trivially holds. Suppose n r β + a i r α β rγ + αβ + N < 0. Rearranging the terms, we obtain a i < N This inequality, together with Condition, implies r α β rγ + αβ +. n r β + r α β rα + αβ + rαβ rβγ + αβ α + n r β + < r α β rγ + αβ + n r β + which simplifies to 0 < β 2 α α rγ β α, a contradiction. nr β+ r α β rγ+αβ+ a i N Therefore 0. Since ln nr β+α+rγ αβ prnγ γ+ β pnα α+ > 0, the above expression describes partner i s unique Nash equilibrium strategy, establishing the last case. 26

27 Proof. Proposition 2 In the previous proof, we established partner i s best response function as the maximum of 0 and prnγ γ+ n ln β pnα α+ σ i s j = a i α + nα β + n rγ + r r α β rγ + αβ + α + nα β + n rγ + r N\i Since r α β rγ+αβ+ 0, if ln prnγ γ+ α+nα β+n rγ+r β pnα α+ 0, partner i s unique optimal strategy is 0, independent of what the other partners choose, making it a strictly dominant strategy for partner i. Alternatively, if α = γ =, the slope of the above expression is zero since r α β rγ + αβ + = r β r + β + = 0. Thus in this case, partner i s best response is again independent of the other partners effort choices, making it a strictly dominant strategy. Therefore, the partnership contract P E [, ] always induces a dominant strategy equilibrium. Note that if γ or α is different than, the slope of the above expression is nonzero, making Partner i s best response dependent on the other partners. Thus, under such partnership contracts, the Nash equilibrium is not a dominant strategy equilibrium. prnγ γ+ Proof. Theorem If ln 0, total effort is 0. Therefore, the statement trivially holds. Alternatively, assume ln > 0. In that case, β pnα α+ total effort is s i = ln r β + a i The derivative of this expression with respect to α is r β+ N ln prnγ γ+ β pnα α+ pr nγ γ + β p nα α + prnγ γ+ β pnα α+ α n nα α + r β + N = < 0. Thus, a decrease in the weight of proportionality in the negative surplus rule increases 6 Note that, this expression gives total effort when α = γ = 0 as well.. 6 s j 27

28 total effort. Next, let us look at the derivative of this expression with respect to n: N n nα α+r β+ n N = nα α + 2 r β + n N nα α + r β + n That is, the above derivative is increasing in absolute value as the number of agents increases. This implies that, in larger partnerships, switching from proportionality to equal shares of negative surplus has a greater effect. Now, let us look at the derivative of total effort respect to γ : prnγ γ+ r β+ N ln β pnα α+ = γ n > 0. nγ γ + r β + N So, as the positive-surplus rule gets closer to proportionality, equilibrium total effort increases. Next, let us check how this derivative changes with n: N = n nγ γ+r β+ n N nγ γ + 2 r β + + n nγ γ + r β + N That is, the above derivative is increasing in absolute value as the number of agents increases. This implies that, in larger partnerships, switching from proportionality to equal shares of positive surplus has a greater effect. Proof. Proposition 4 Introducing the equilibrium effort into partner i s utility function, we obtain U P E[γ,α] i s = p p prnγ γ+ β pnα α+ prnγ γ+ β pnα α+ rγ α+rγ αβ pr nγ γ + β p nα α + 28 α β α+rγ αβ n < 0. > 0. ra i βγ α nr β+α+rγ αβ