Social Norms for Team Work

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1 Social orms for Team Work Marc-Olivier Moisan-Plante February 7, 003

2 Introduction In his seminal 198 paper, Hölmstrom showed that there do not exist sharing rules among a partnership that yields the first-best outcome when members of a team cannot observe the effort of their colleagues. This has been referred to be the 1 problem in the literature and it is a case of free-riding. Several alternatives were developed 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 implementation through mechanism design (i.e. Sharma and Torres [001]). Another approach taken was thnvestigation of the effects of peer pressurn the team (an early exampls Kandel and Lazear [199]). The pressure can come from the psychological cost (i.e. guilt) of a downward deviation of effort relative to a social norm or bnstituted 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 effort 1. The next section presents a model adapted from Huck, Kübler and Weibull [00] in which thers 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

3 1 A Model of Peer Pressure The model is essentially the one presented in section of Huck, Kübler and Weibull [00]. The basic structurs based on the work of Holmström [198]. There are symmetric players in a team. The production function is: Y = where the effort level of member i is: [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 technology. 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. The disutility from exerting effort is growing quadratically in its level: i=1 e 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 social norm is assumed to be quadratic: δ ( x ) where δ is a sensibility parameter. 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). Lastly, we consider the sharing rule to be 1 here and elsewhern the paper. Although it is not sophiticated, it is quite natural and likely to be adopted in practice For examplf the production function takes the form of Y = α and effort cost is, the ash equilibrium level of effort will be e = α 1. If we let α = represents varying returns to scale, the utility of members will bmized at = 106 with e 0.97 (the effort level would bmized at e 1.73 for = 4). i=1

4 among symmetric partners. The problem facing each partner is therefore: + e j δ ( x ) subject to the non-negativity constraint on. The optimal level of effort is found by taking the first-order condition: e = δ x 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 approaches zero, the effort level collapses to 1 (i.e. the 1 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 imizing with respect to x 3. That is: x Straightforward computations yields: e e δ ( x e ) x = 1 3 Symmetry implies that imizing a member welfars equivalent to imizing social welfare. (1) () 3

5 The optimal social norm equals the first-best 4. The result seems intuitive as the social norm reflects what the effort level should bn 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 (i.e. δ( x )), we would have had another reaction function (i.e. e = δ + 1 ) and another optimal team objective (i.e. x = δ + 1 ). The arbitrariness of the social cost function (hence the social norm) seems deceptive 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 ust in more elaborated models. This will be done next where costly observability (i.e. monitoring) is required prior exerting peer pressure on a member. A Model with Monitoring Wntroducn this section thdea that effort observability is not free. More specifically we suggest that applying pressure on a peer requires prior costly monitoring of its effort level. We denote by a i [0, ) the monitoring level chosen by agent i. The cost incurred by agent i are given by θa i. The parameter θ can be though to be much smaller than 1 to express thdea that monitoring is less costly than effort in general 5. We denote by a i the average monitoring level of the 1 agents j i that is exerted on agent i. That is: a i = a j 1. The cost function associated with the social norm is: δ ( x ) a i and therefore reflects that higher monitoring involve a greater 4 By first-best level of effort we denote the level of effort that would imize social welfare when there are no free-rider problem, i.e. if each member would receivts individual contribution instead of the team average contribution. The first-best level of effort solves: e i 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(, a i ) are possible at the expense of increased complexity.. 4

6 punishment for those who deviate from the social norm. It is implicitly assumed that the monitoring is symmetric (the monitoring effort a i of member i is equally divided on the 1 other agents). We maintain a 1 sharing rule. This can be justified by the fact that effort is not contractible (although observable at some costs) so that monetary punishments arnfeasible. 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 stags therefore: a i + e j θa i δ ( x ) a i (3) subject to the non-negativity constraint on a i. In the second stage the agents problem is: + e j subject to the non-negativity constraint on. The problem is solvable by backward induction. θa i δ ( x ) a i (4) As the agents are solving a problem of nested imization, this will insure subgame perfection. From the last stage we get the effort reaction function: ê i = 1 + δ xa i 1 + δa i (5) which is substituted back for every agent in the first stage objective function: a i ê i + ê j ê i θa i δ ( x ê i) a i (6) Here, when choosing an optimal monitoring level a i, agent i must compute the effect of a i on the 1 other partners effort reaction functions ê j as a i a j. 5

7 Simultaneously solving each member first order condition 6 yields in a symmetric equilibrium: a = x 1 δθ The equilibrium level of effort is therefore: θ e = x x 1 (8) δ A few remarks about those equations: First, for the model to bnteresting we must have the parametric restriction: x > θ + 1. This insures a positive monitoring level. This is quitntuitive as 1 δ is the equilibrium effort level when no peer pressure possibility exists. A meaningful social norm must be greater. ote that when x = θ + 1 the optimal monitoring δ level is zero and the effort level collapses to 1 7. 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 ( x 1) to imize welfare. The same logic goes 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 ( x 1) to imize partners welfare. 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 This is also the case for social norms lower than θ δ + 1. It could be verified using Kuhn-Tucker multipliers. 6 1 δ (7)

8 consider increasing returns in the production, as mentioned earlier, thntroduction 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,. As all agents are alike, this is the same as to imize a partner utility in equilibrium: x e e θa δ ( x e ) a (9) As mentioned earlier, thmization over x could be though to take place during 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. The parameters will be = 10, θ = 1, δ = 10. The parametric restriction 10 is: x >. On the following figure, utility, effort and monitoring levels are plotted 10 against the level of the social norm x. The curve on the top is the effort level, the onn the middls the monitoring and the last ons utility. As we can see, effort and monitoring arncreasing in x. Mornterestingly the utility curvs concave (at least over the region considered) and has an apparently uniqumum 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 device. When social expectations are high, the peer pressurs greater for a given level 7

9 Figure 1: Effort, Utility Monitoring on the vertical axis, x on the vertical axis. of effort (recall that the peer pressurncreases quadratically in the gap between realized 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 hncurs 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 pressurs 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 ( x 1) as either θ 0 or δ we can conjecture that the 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

10 Partial analytical results provide support for this hypothesis. As shown in the appendix 8 we are able to claim that the optimal social norm does not equal the firstbest for teams with more than three members 9. When = 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 specializn 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 specializn monitoring he will be the only one to incur those costs as the other partners will specializn work effort. Including those fixed cost the symmetric problem becomes: a i, + e j θa i ψ δ ( x ) a i (10) 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. ow consider the problem of a worker specializing in work effort in a asymmetric task assignment. Assume that in a team of 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

11 M workers and define the monitor to worker ratio; λ = problem is to solve: + e j M 1 M. The worker s M δ ( x ) λa (11) subject to the non-negativity constraint on. ote that here, a is the monitoring intensity chosen by one monitor. Solving for e yield the familiar looking equation: ê i = 1 + δλ xa 1 + δλa (1) At first sight effort per worker will diminish. However we will solve the monitor s problem: a i M e j θa i ψ (13) The monitoring intensity a i of monitor i enters the reaction function of each of the M workers. Solving, we get: a = x 1 λ δθ 1 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 λ s cancel out when substituting a in ê i ). However thers 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, = 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 thers only one monitor and the weights are their relative sharn the team. The monitor utility curvncreases monotonically 10 (14)

12 Figure : 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. Thnteresting thing to see hers the potential conflict between workers and monitors. If wmize team welfare, we have to have a team objective greater than what would imize 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 objectivncreases the expected punishment that will be exerted on workers while caught shirking. Hence workers will work harder which benefit to the monitor. Moreover, the monitor can adjust its monitoring intensity for further utility gains. This suggests that the preplay negotiations might not be easy. Thers clearly a rent to be a monitor in this case 10. 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

13 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. 1

14 Conclusion We have looked at a sequential mutual monitoring gamnvolving a social norm for work effort under a specific parameterization. We solved the subgame perfect equilibrium 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 important 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, wnvestigated how would the two groups, workers and monitors, prefer to set the team objective. Thers 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 equilibrium. 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 commitment 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 pursun 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 unresolved. A production function displaying varying return to scale could affect the analysis importantly. 13

15 Bibliography Banker R. and Lee S.Y., (undated) Mutual Monitoring and Peer Pressurn Teamwork, Working Paper. Barron J. and Gjerde K. (1997) Peer Pressurn a Agency Relationship, Journal of Labor Economics 15, Holmström B., (198) Moral Hazard in Teams, The Bell Journal of Economics 13, Huck S., Kübler D. and Weibull J., (00) Social orms and Incentives in Firms, Working Paper. Huddart S. and Liang P.J., (00) Profit Sharing in Partnerships, Working Paper. Kandel E. and Lazear E., (199) Peer Pressure and Partnerships, The Journal of Political Economy 100, Legros P. and Matthews S., (1993) Efficient and early Efficient Partnerships, The Review of Economic Studies 60, Radner R., (1986) Repeated Partnership Games with Imperfect Monitoring and o Discounting, The Review of Economic Studies 53, Sharma T. and Torres J., (001) Coordination in Teams, Working Paper. 14

16 Appendix The equilibrium monitoring level is given by: a i ê i + ê j ê i θa i δ ( x ê i) As effort level ê j of agents j i depends on the monitoring level a i chosen by agent i we need to keep track of the ê j s. Thmization can be rewritten keeping only the relevant terms: a i ê j θa i Substituting the definition of ê in the last expression and using thndex k for k i and k j, agent i problem s is: ( ai+ 1 + δ x a i ( ai+ 1 + δ a k k 1 a k k 1 Expanding and summing over j and k we get: a i 1 ) ) ( 1 + δ xa i + δ x( )a k δa i + δ( )a k 1 1 Taking the first-order with respect to a i we get: ( δ x ) ( ) θ 1 + δa i + δ( )a k 1 = δ 1 ( ( 1 + δa i + δ( )a k 1 1 j θa i ) a i θa i 1 + δ xa i + δ x( )a k 1 1 ) We now impose symmetry in equilibrium: a i = a k, multiply both sides by 1, develop the numerator and regroup terms in the denominator to get: ) θ = δ ( x 1 (1 + δa) This is an equation of the second degren 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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