Docstoc

Social Norms for Team Work

Document Sample
Social Norms for Team Work Powered By Docstoc
					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