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Social Norms for Team Work Marc-Olivier Moisan-Plante February 27, 2003 Introduction o In his seminal 1982 paper, H¨lmstrom showed that there do not exist sharing rules among a partnership that yields the ﬁrst-best outcome when members of a team 1 cannot observe the eﬀort of their colleagues . This has been referred to be the ” N problem” in the literature and it is a case of free-riding. Several alternatives were de- veloped to circumvent the problem, among them repeated game settings (i.e. Radner [1986]), the use of mixed strategies (i.e. Legros and Matthews [1993]) or implemen- tation through mechanism design (i.e. Sharma and Torres [2001]). Another approach taken was the investigation of the eﬀects of peer pressure in the team (an early example is Kandel and Lazear [1992]). The pressure can come from the psychological cost (i.e. guilt) of a downward deviation of eﬀort relative to a social norm or be instituted by peers by social ostracism (see Banker and Lee [undated] for a more complete discussion and their references). Also when the peer pressure involve costs for the pressurizers (the observability of other members eﬀort level’s involve costs) the term (mutual) ”monitoring” is also used. We will not attempt to review this literature here as a comprehensive survey would constitute another paper by its own. Rather, in a context of a partnership where the monitoring and eﬀort are chosen sequentially, we will try to characterize what would be the optimal level of a social norm regarding work eﬀort1 . u The next section presents a model adapted from Huck, K¨bler and Weibull [2002] in which there is peer pressure but no monitoring. The section two introduces a model with monitoring and peer pressure. Section three considers this latter model in the context of asymmetric task assignments. Finally the conclusion will discuss possible improvements and extensions. 1 In this aspect we specialize the general model from Barron and Gjerde [1997] to incorporate an endogenously chosen social norm. 1 1 A Model of Peer Pressure u The model is essentially the one presented in section 2 of Huck, K¨bler and Weibull o [2002]. The basic structure is based on the work of Holmstr¨m [1982]. There are N N symmetric players in a team. The production function is: Y = ei where the i=1 eﬀort level of member i is: ei ∈ [0, ∞). We note that this linearly additive production function do not capture the essence of the partnership; there are probably increasing returns over some range of the size of the partnership to justify the creation of a team. The size of the ﬁrm should be determined by the production technology2 . However the linearly additive production function is analytically simple and convenient to express the free-rider problem and it is often used in this context . e2 The disutility from exerting eﬀort is growing quadratically in its level: 2 i . We ¯ assume the existence of a social norm x that dictates the socially accepted level of eﬀort by a member of the team. We assume that a worker deviating from the social norm incurs a cost due to social ostracism (this includes all possible forms of peer pressure: psychological harassment, physical harassment, shame, public humiliation etc.). The cost to deviate from the 2 social norm is assumed to be quadratic: δ (¯−ei ) where δ is a sensibility parameter. x 2 It is implicitly assumed that the eﬀort level of a member of the team is observable at no cost to the other members (costly observability is introduced in the next section). 1 Lastly, we consider the sharing rule to be ” N ” here and elsewhere in the paper. Although it is not sophiticated, it is quite natural and likely to be adopted in practice N 2 e2 For example if the production function takes the form of Y = N α ei and eﬀort cost is 2 , i i=1 N the Nash equilibrium level of eﬀort will be e = N α−1 . If we let α = 1.1 − 1000 represents varying returns to scale, the utility of members will be maximized at N = 106 with e 0.972 (the eﬀort level would be maximized at e 1.273 for N = 24). 2 among symmetric partners. The problem facing each partner is therefore: ej ei j=i e2 (¯ − ei )2 x max + − i −δ (1) ei N N 2 2 subject to the non-negativity constraint on ei . The optimal level of eﬀort is found by taking the ﬁrst-order condition: x 1 δ¯ + N e = 1+δ We can see that as the partner get more sensible to peer pressure (i.e. δ gets large), ¯ the level of eﬀort tends to approach the social norm x. Also the greater is the social norm, the greater is the eﬀort exerted. Finally as the sensibility to peer pressure 1 1 approaches zero, the eﬀort level collapses to ” N ” (i.e. the ” N problem”): a partner do not want to work hard, as he expects the others to free-ride on his eﬀort (the output is shared evenly among all members but the costs of eﬀort are borne on an individual basis). What is the level of the social norm ? If we talk about a ”social norm” it seems to be the case that it comes from a long and slow evolutive process. However, given that enough time has elapsed for far, this process should have come through an end and reached optimality. Also, if we consider the social norm to be a team objective set during preplay negotiations among the partners, we should again expect this objective to be chosen optimally as all players are symmetric which annihilates possible conﬂicts of interests. Hence, the social norm (or team objective) can be found substituting optimal eﬀort e in a partner objective function and maximizing with respect to x 3 . That is: ¯ e2 (¯ − e )2 x max e − −δ (2) ¯ x 2 2 Straightforward computations yields: ¯ x =1 3 Symmetry implies that maximizing a member welfare is equivalent to maximizing social welfare. 3 The optimal social norm equals the ”ﬁrst-best”4 . The result seems intuitive as the social norm reﬂects what the eﬀort level should be in a world without free-ridering, but actually it is not quite so. In fact this result is driven by the underlying mathematical structure of the social cost function. Had we speciﬁed another social cost function 1 (i.e. δ(¯ − ei )), we would have had another reaction function (i.e. e = δ + x N ) and 1 ¯ another optimal team objective (i.e. x = δ + N ). The arbitrariness of the social cost function (hence the social norm) seems decep- tive at ﬁrst sight. However we note that the cost function used by Huck and al. has the ”focal” property to induce the ﬁrst-best level of eﬀort in this simple example and permits easier comparisons, serving as a yardstick, when we shall use it in more elab- orated models. This will be done next where costly observability (i.e. monitoring) is required prior exerting peer pressure on a member. 2 A Model with Monitoring We introduce in this section the idea that eﬀort observability is not free. More specif- ically we suggest that applying pressure on a peer requires prior costly monitoring of its eﬀort level. We denote by ai ∈ [0, ∞) the monitoring level chosen by agent i. The cost incurred by agent i are given by θai . The parameter θ can be though to be much smaller than 1 to express the idea that monitoring is less costly than eﬀort in general5 . We denote by a−i the average monitoring level of the N − 1 agents j = i that is aj j=i exerted on agent i. That is: a−i = N −1 . The cost function associated with the social )2 norm is: δ (¯−ei a−i and therefore reﬂects that higher monitoring involve a greater x 2 4 By ”ﬁrst-best” level of eﬀort we denote the level of eﬀort that would maximize social welfare when there are no free-rider problem, i.e. if each member would receive its individual contribution e2 instead of the team average contribution. The ”ﬁrst-best” level of eﬀort solves: max ei − 2 . i ei 5 We note that the use of a separable cost function in eﬀort and monitoring involve a loss of generality. More general costs functions, C(ei , ai ) are possible at the expense of increased complexity. 4 punishment for those who deviate from the social norm. It is implicitly assumed that the monitoring is symmetric (the monitoring eﬀort ai of member i is equally divided on the N − 1 other agents). 1 We maintain a ” N ” sharing rule. This can be justiﬁed by the fact that eﬀort is not contractible (although observable at some costs) so that monetary punishments are infeasible. Finally the timing is as follows: In the ﬁrst stage agents decide of their monitoring level and in the second stage they chose their eﬀort level and payoﬀs are realized. Agent ”i” problem’s in the ﬁrst stage is therefore: ej ei j=i e2 (¯ − ei )2 x max + − i − θai − δ a−i (3) ai N N 2 2 subject to the non-negativity constraint on ai . In the second stage the agents problem is: ej ei j=i e2 (¯ − ei )2 x max + − i − θai − δ a−i (4) ei N N 2 2 subject to the non-negativity constraint on ei . The problem is solvable by backward induction. As the agents are solving a problem of nested maximization, this will insure subgame perfection. From the last stage we get the eﬀort reaction function: 1 N + δ¯a−i x ˆ ei = (5) 1 + δa−i which is substituted back for every agent in the ﬁrst stage objective function: ˆ ej ˆ ei j=i e2 ˆ (¯ − ei )2 x ˆ max + − i − θai − δ a−i (6) ai N N 2 2 Here, when choosing an optimal monitoring level ai , agent ”i” must compute the ˆ eﬀect of ai on the N − 1 other partners eﬀort reaction functions ej as ai ∈ a−j . 5 Simultaneously solving each member ﬁrst order condition6 yields in a symmetric equilibrium: ¯ 1 x− N 1 a = √ − (7) N δθ δ The equilibrium level of eﬀort is therefore: θ√ ¯ e =x− ¯ Nx − 1 (8) δ A few remarks about those equations: First, for the model to be interesting we must have the parametric restriction: θN 1 1 ¯ x> δ + N . This insures a positive monitoring level. This is quite intuitive as N is the equilibrium eﬀort level when no peer pressure possibility exists. A meaningful θN 1 ¯ social norm must be greater. Note that when x = δ + N the optimal monitoring 1 7 level is zero and the eﬀort level collapses to N . As the cost of monitoring θ increases, the social norm lower bound for positive monitoring must be higher. Larger teams and low sensibility to peer pressure (i.e. low δ) are also causing the social norm lower bound to increase. As the monitoring technology becomes increasingly eﬃcient (θ −→ 0) the optimal ¯ eﬀort level approaches the social norm (e −→ x) as a lot of monitoring is undertaken so that shirking becomes increasingly costly. In this case the optimal social norm should tend to the ﬁrst best (¯ −→ 1) to maximize welfare. The same logic goes x through when partners get more sensible to peer pressure (as δ −→ ∞ then e −→ ¯ x) and the optimal social norm should again approach the ﬁrst-best level of eﬀort (¯ −→ 1) to maximize partners welfare. x Finally as the size of the team increase, for given θ and δ, agents shirk increasingly on both eﬀort and monitoring (they do not get full returns on either task). Unless we 6 The derivation is given in the appendix. 7 θN 1 This is also the case for social norms lower than δ + N . It could be veriﬁed using Kuhn-Tucker multipliers. 6 consider increasing returns in the production, as mentioned earlier, the introduction of a peer pressure environment can only diminish the free-rider problem, not to remove it. We now investigate the level of the optimal social norm for intermediate parameter values (satisfying the previous parametric restriction) . We are looking for is an ¯ optimal social norm x as a function of the parameters of the model θ, δ, N . As all agents are alike, this is the same as to maximize a partner utility in equilibrium: 2 e (¯ − e )2 x max e − − θa − δ a (9) ¯ x 2 2 ¯ As mentioned earlier, the maximization over x could be though to take place dur- ing preplay negotiation between members of the team. As members are homogenous, ¯ the agreement over x should be reached swiftly. ¯ Unfortunately, as e and a are themselves rather complicated function of x (recall equations 7 and 8) we will not provide a closed form solution for this exercise. Instead, we will ﬁrst proceed by graphical inspection for reasonable parameters values before providing some partial analytical results. 1 The parameters will be N = 10, θ = 10 , δ = 10. The parametric restriction 2 ¯ is: x > 10 . On the following ﬁgure, utility, eﬀort and monitoring levels are plotted ¯ against the level of the social norm x. The curve on the top is the eﬀort level, the one in the middle is the monitoring ¯ and the last one is utility. As we can see, eﬀort and monitoring are increasing in x. More interestingly the utility curve is concave (at least over the region considered) ¯ and has an apparently unique maximum x > 1. The optimal social norm of work eﬀort exceeds the ﬁrst-best level. In equilibrium partners expects more eﬀort from peers than what would be optimal without a monitoring technology, peer pressure environment and free-riding problem. However with monitoring technology and peer pressure environment, partners prefer having high expectations as a commitment de- vice. When social expectations are high, the peer pressure is greater for a given level 7 ¯ Figure 1: Eﬀort, Utility Monitoring on the vertical axis, x on the vertical axis. of eﬀort (recall that the peer pressure increases quadratically in the gap between real- ized eﬀort and the norm). From a partner point of view this induces more eﬀort from his colleagues and ultimately increases his well-being despite the disutility he incurs through rampant peer pressure and monitoring costs in equilibrium. This creates an endogenously determined ”stressful” work environment. Everybody expects a lot of work from other members and although nobody meet the standard of eﬀort, the work eﬀort under peer pressure is increased relative to an environment without peers control. Using our previous observation that the optimal social norm tends to the ﬁrst-best level of eﬀort (¯ −→ 1) as either θ −→ 0 or δ −→ ∞ we can conjecture that the x optimal social norm approaches 1 from the right. That is, the better is the monitoring technology (i.e. the lower is θ) or the more sensible partners are to peer pressure (i.e. the higher is δ) the lower the optimal social norm should be. 8 Partial analytical results provide support for this hypothesis. As shown in the appendix8 we are able to claim that the optimal social norm does not equal the ﬁrst- best for teams with more than three members9 . When N = 3, the optimal work standard is the ﬁrst-best, independently of the parameters values (given that they satisfy the parametric restriction). 3 A Model with Asymmetric Tasks Up to now we have assumed that the members had symmetric task, that is they all work and monitor in the same way. However, it is not too diﬃcult to imagine that when the team gets large there could be gain to specialization in the diﬀerent tasks. For example, if one member specialize in monitoring, he might get more eﬃcient at supervising his colleagues as he will learn more about how to verify correctly their work eﬀort without being fooled by them. An easy way to incorporate this in the model, is to introduce ﬁxed costs in the monitoring activity. Those costs can be though to represent time spent learning diﬀerent abilities to monitor properly. If one partner specialize in monitoring he will be the only one to incur those costs as the other partners will specialize in work eﬀort. Including those ﬁxed cost the symmetric problem becomes: ej ei j=i e2 (¯ − ei )2 x max + − i − θai − ψ − δ a−i (10) ai , e i N N 2 2 where ψ are the ﬁxed costs associated with monitoring. As those costs drop out when we take the ﬁrst order conditions, they do not change the decisions rules found earlier, but constitute a utility downshift. Now consider the problem of a worker specializing in work eﬀort in a asymmetric task assignment. Assume that in a team of N members, there are M monitors and 8 TO DO. 9 In fact graphical inspection suggests that it is also strictly greater. Also for teams of two members, the optimal social norm seems to be lower than one. 9 M N − M workers and deﬁne the monitor to worker ratio; λ = N −M . The worker’s problem is to solve: ej ei M −N −1 e2 i (¯ − ei )2 x max + − −δ λa (11) ei N N 2 2 subject to the non-negativity constraint on ei . Note that here, a is the monitoring intensity chosen by one monitor. Solving for e yield the familiar looking equation: 1 N + δλ¯a x ˆ ei = (12) 1 + δλa At ﬁrst sight eﬀort per worker will diminish. However we will solve the monitor’s problem: ej M max − θai − ψ (13) ai N The monitoring intensity ai of monitor i enters the reaction function of each of the N − M workers. Solving, we get: ¯ 1 x− N 1 a = √ − (14) λ N δθ λδ The higher optimal monitoring intensity chosen by each monitor just compensate the fact that there are only a few of them so that eﬀort per worker stays the same (The ˆ λ’s cancel out when substituting a in ei ). However there is now a deﬁcit of workers (hence output). As a compensation workers do not have to spent eﬀort on monitoring avoiding the ﬁxed costs and the monitoring costs while monitors do not have to produce output. If the ﬁxed costs are high enough so that the task specialization is worthwhile, everybody can be better oﬀ, as it is the case when M = 1, ψ = 0.1 and again θ = 0.1, N = 10, and δ = 10. The results are shown in the following ﬁgure: As we can see the worker’s utility curve lies above the symmetric utility curve and drives the weighted utility curve as there is only one monitor and the weights are their relative share in the team. The monitor utility curve increases monotonically 10 ¯ Figure 2: Utility on the vertical axis, social norm x on the vertical axis. with the level of the social norm and goes above the symmetric utility curve for a norm high enough. The interesting thing to see here is the potential conﬂict between workers and monitors. If we maximize ”team welfare”, we have to have a team objective greater than what would maximize worker’s welfare (i.e. the ”team optimum” is around ¯ ¯ x = 1.07, but the worker’s preferred social norm is around x = 0.98). This is due to the fact that monitor’s utility increases monotonically with the level of the team objective. For a monitor (keeping its monitoring intensity constant), an higher team objective increases the expected punishment that will be exerted on workers while caught shirking. Hence workers will work harder which beneﬁt to the monitor. More- over, the monitor can adjust its monitoring intensity for further utility gains. This suggests that the preplay negotiations might not be easy. There is clearly a rent to be a monitor in this case10 . In our parameterized example, the workers prefer 10 It is also possible that monitors be worse oﬀ the symmetric case while the workers be better oﬀ. 11 a team objective below the ﬁrst-best, while the monitors prefer to have one higher than the ﬁrst-best. While this do not constitute a sharp prediction, it illustrate the fact that a social norm or team objective might be the outcome of some bargaining game prior the structure of the team is established. However, by increasing the number of monitor, we can generally fall back to our case. 12 Conclusion We have looked at a sequential mutual monitoring game involving a social norm for work eﬀort under a speciﬁc parameterization. We solved the subgame perfect equilib- rium of the game and analyzed how should an optimal social norm for work eﬀort be determined. We got the surprising result that it should overshoot the ﬁrst-best level of eﬀort. Partners use the monitoring technology and peer pressure environment to commit to eﬀort optimally. Even so in equilibrium free-riding eﬀects are still impor- tant as the realized work eﬀort fall short of the norm. Partners essentially trade-oﬀ disutility from peer pressure and monitoring costs against higher collective eﬀort. Allowing for task specialization, we investigated how would the two groups, work- ers and monitors, prefer to set the team objective. There is a clear conﬂict of interest between them. Monitors always prefer higher norms than workers. This will lead to a diﬃcult bargaining process to establish the team structure among otherwise totally symmetric agents. The sequentiality of the model permits to obtain positive monitoring in equilib- rium. Partners cannot revise their monitoring decision in the second stage. Allowing simultaneous decisions for eﬀort and monitoring would drive the monitoring level down to zero. With the multi-stage game, members beneﬁt from a perfect com- mitment technology with respect to monitoring decisions. As noted by Barron and Gjerde [1997] a repeated game setting would perhaps justify this apparent easiness to commit. Also, if we were to pursue in this way, we would need to consider how does the social norm evolves from period to period. Finally, the problem of a consistent production function for teams is still unre- solved. A production function displaying varying return to scale could aﬀect the analysis importantly. 13 Bibliography Banker R. and Lee S.Y., (undated) ”Mutual Monitoring and Peer Pressure in Team- work,” Working Paper. Barron J. and Gjerde K. (1997) ”Peer Pressure in a Agency Relationship,” Journal of Labor Economics 15,234-254. o Holmstr¨m B., (1982) ”Moral Hazard in Teams,” The Bell Journal of Economics 13, 324-340. u Huck S., K¨bler D. and Weibull J., (2002) ”Social Norms and Incentives in Firms,” Working Paper. Huddart S. and Liang P.J., (2002) ”Proﬁt Sharing in Partnerships,” Working Paper. Kandel E. and Lazear E., (1992) ”Peer Pressure and Partnerships,” The Journal of Political Economy 100, 801-817. Legros P. and Matthews S., (1993) ”Eﬃcient and Nearly Eﬃcient Partnerships,” The Review of Economic Studies 60, 599-611. Radner R., (1986) ”Repeated Partnership Games with Imperfect Monitoring and No Discounting,” The Review of Economic Studies 53, 43-57. Sharma T. and Torres J., (2001) ”Coordination in Teams,” Working Paper. 14 Appendix The equilibrium monitoring level is given by: N ˆ ej ˆ ei j=i e2 ˆ (¯ − ei )2 x ˆ max + − i − θai − δ a−i ai N N 2 2 ˆ As eﬀort level ej of agents j = i depends on the monitoring level ai chosen by agent ˆ i we need to keep track of the ej ’s. The maximization can be rewritten keeping only the relevant terms: ˆ ej j=i max − θai ai N ˆ Substituting the deﬁnition of ej=i in the last expression and using the index k for k = i and k = j, agent i problem’s is: ai + ak 1 k N + δ¯x N −1 j=i max ai + ak − θai ai N 1 N +δ N −1 k j Expanding and summing over j and k we get: 1 ¯ δ xai ¯ δ x(N −2)ak N −1 N + N −1 + N −1 max δ(N −2)ak − θai ai N 1+ δai + N −1 N −1 Taking the ﬁrst-order with respect to ai we get: ¯ δx δai δ(N −2)ak δ 1 ¯ δ x ai δ x(N −2)ak ¯ θN N −1 1+ N −1 + N −1 − N −1 N + N −1 + N −1 = 2 N −1 δai δ(N −2)ak 1+ N −1 + N −1 We now impose symmetry in equilibrium: ai = ak , multiply both sides by N − 1, develop the numerator and regroup terms in the denominator to get: 1 ¯ δ x− N θN = (1 + δa)2 This is an equation of the second degree in a. Using the quadratic formula and keeping only the ”+” root we obtained after further algebraic manipulations the equation given in the text. 15

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