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Social norms and incentives in firms Steffen Huck_ Dorothea Kübler

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Social norms and incentives in firms Steffen Huck_ Dorothea Kübler Powered By Docstoc
					                        Social norms and incentives in firms
           Steffen Huck, Dorothea Kubler and Jorgen Weibull∗
                                   ¨          ¨

                   4 May 2002. Revised version of IUI WP 565, 2001.


          Abstract. This paper studies the interplay between economic incentives
       and social work norms in firms. We outline a simple model of team production
       and provide results for linear incentive schemes in the presence of a social norm
       that may cause multiple equilibria.

          JEL code: D23
          Keywords: social norms, incentives, contracts.



                                1. Introduction
Behavior in firms is most likely not only governed by economic incentives but also by
social norms among the employees. This paper deals with the interplay of these two
forces. Indeed, nowadays it is not unusual to include social norms in microeconomic
analyses.1 However, there have yet not been many attempts at studying how social
norms affect the incentive structure within firms.2 This note outlines a new attempt
in this direction.
    For a firm owner, social norms concerning work are important because they can
affect profits. For example, norms may influence how much effort workers put into
projects where only joint output is observable. Social norms, emanating from the
ratchet effect, may also keep workers from working hard under relative-performance
   ∗
      Huck is at University College London, Department of Economics & ELSE, Gower Street, London
                                    u
WC1E 6BT, United Kingdom; K¨bler at Humboldt University Berlin, Department of Economics,
Spandauer Str. 1, D-10178 Berlin, Germany; and Weibull at the Stockholm School of Economics,
Department of Economics, P.O. Box 6501, SE - 113 83 Stockholm, Sweden, and at the Research
Institute of Industrial Economics, P.O. Box 5501, SE - 114 85 Stockholm, Sweden. We are grateful
for comments from Tore Ellingsen and from participants in a seminar at the Stockholm Institute for
Transition Economics. This note is an elaboration of the rough sketch in Weibull (1997).
    1
      See e.g. Akerlof (1980), Moffitt (1983), Besley and Coate (1992), Bernheim (1994), Lindbeck,
                                                   u
Nyberg, and Weibull (1999), Hart (2001), and K¨bler (2001), and the literature cited in these
studies.
    2
      See Kandel and Lazear (1992) and Barron and Gjerde (1997). Also Hart (2001) focuses on
norms and firms, but rather deals with the question whether the degree of trust between agents
influences the optimal ownership structure.

                                                1
                             Social norms and incentives in firms                        2


schemes or piece-rate schemes that are adjusted according to past performance. Under
many such contracts, the compensation to a worker not only depends on his or her
own effort level, but also on the effort of other workers. Moreover, peer pressure
penalizes those who deviate from the group norm, and depending on that norm,
output may be higher or lower than without the norm. On top of this, social norms
may cause multiplicity of equilibria associated with an incentive scheme.
    We analyze work norms in a static model of team production.3 Each individual
agent’s effort level is unobserved by the principal, but total output can be observed
and verified. The principal chooses among linear incentive schemes in order to maxi-
mize profits, and the agents in the team simultaneously choose their efforts thereafter.
We study the effect of a social norm concerning work effort among the team mem-
bers. In particular, we show how the optimal incentive scheme, in the class of linear
schemes, depends on the social norm.
    We believe that the possibility of multiple equilibria, which such models allow
for, has empirical relevance. For instance, Ichino and Maggi (2000) find substantial
shirking differentials between branches of an Italian firm, despite identical monetary
incentives governing the employees’ efforts in these branches. They identify group-
interaction effects as a key explanatory variable that allows for multiple equilibria.
However, they do not analyze the interplay of these group-interaction effects with
economic incentives. Encinosa, Gaynor, and Rebitzer (1997) find that group norms
matter in medical partnerships. Their focus is on the interplay between group norms,
multi-tasking and risk aversion.
    Whether multiple equilibria exist in theory depends on agents’ social preferences.
While Kandel and Lazear (1992) rule out multiplicity by assuming that peer pressure
meets certain regularity conditions, we side with Lindbeck et al. and argue that
such restrictions are hard to justify a priori. If the regularity conditions are violated,
then equilibria with low efforts and low social pressure can coexist, under the same
incentive scheme, with equilibria with high efforts and high social pressure. This
multiplicity is relevant for a principal who strives to find a profit-maximizing incentive
scheme. For example, a firm trapped in a low-effort equilibrium may “jump up” to a
high-effort equilibrium even by way of a small increase in the bonus if the equilibrium
correspondence has a fold just above the current bonus rate, and likewise for sudden
downward jumps (see section 3). From a dynamic perspective, it may be beneficial for
firm owners to temporarily change the economic incentive scheme, or in some other
way temporarily inspire the workers to high efforts (even at a high temporary cost)
in order to lead the team to the basin of attraction of a high-effort equilibrium. Such
”jump starting” may lead the team away from a low-effort equilibrium. Afterwards,
  3
      For a pioneering model of team production, see Holmstrom (1982).
                            Social norms and incentives in firms                               3


the temporary stimulus may be withdrawn while the workers’ efforts remain at the
desired high level, due to the new and more demanding work norm.4 In our simple
model, multiple equilibria are Pareto ranked according to effort, so also the workers
prefer the high-effort (and thus high profit) equilibrium.
    The paper is organized as follows. The basic is introduced in section 2, where we
start by outlining the benchmark case of no social norm, and then point out some of
the model’s general qualitative properties. These general observations are illustrated
in some detail in an example in section 3. Section 4 discusses briefly some variations
and generalizations of the model assumptions. Section 5 concludes.

                                    2. The model
We consider team production with a profit maximizing owner (the principal) as resid-
ual claimant. There are n > 1 identical workers (agents). Each worker i exerts
some effort xi ≥ 0. Let x−i denote the average effort exerted by all other workers,
       P
x−i = j6=i xj / (n − 1). The production technology is linear: output y equals the
                                  P
sum of all workers’ efforts, y = n xi . The principal can only observe aggregate
                                      i=1
output y, not individual efforts. The workers are also assumed to observe aggregate
effort, from which each worker i can deduce the average effort x−i of the others (as-
suming knowledge of the production technology and his or her own effort). In order
to focus on the interplay between economic incentives and social norms in the sim-
plest possible setting, this study is restricted to linear contracts, which are common
in practice and allow for a transparent analysis. More precisely, each worker earns
the same wage w, and this wage is an affine function of the firm’s output,

                                       w = a + by/n .

where the owner chooses the fixed salary a and the bonus rate b. We require a to be
nonnegative, an assumption which can be justified by wealth constraints.5 Therefore,
the profit-maximizing owner will optimally choose b in the open unit interval, 0 <
b < 1.
   Assuming that the firm is a price taker in its product market, and normalizing
the market price to unity, the firm’s profit – the residual left to the owner – is thus

                                     π = (1 − b)y − na.
   4
     See Lindbeck, Nyberg, and Weibull (1999) for a similar argument in the context of transfers
and taxes in a welfare state.
   5
     Otherwise the principal could sell the firm to each of the workers. In the absence of social
norms, such an arrangement would make the first-best solution achievable. However, such schemes
are vulnerable to collusion and sabotage.
                           Social norms and incentives in firms                             4


2.1. Without a group norm. We first analyze the benchmark case when social
norms are absent or have no influence on behavior. A worker’s utility then only
depends on his or her wage earning and effort. We assume throughout this study
that the workers are identical and have linear-quadratic utility in consumption and
work-effort:
                            1        b                     1
                    ui = w − x2 = a + [(n − 1) x−i + xi ] − x2 .
                              i
                            2        n                     2 i
   From this it is immediate that workers’ decisions concerning effort are strategically
independent. Regardless of whether workers decide simultaneously or sequentially,
each worker solves
                                           b 1
                                     max xi − x2 .
                                     xi ≥0 n 2 i
Consequently, the unique equilibrium effort level, given any contract (a, b), are xi =
b/n for all workers i.
    Viewing workers’ effort choices as a subgame following upon the owner’s choice
of contract, we solve the full game for subgame perfect equilibrium by inserting the
above (subgame equilibrium) effort levels into the expression for the firm’s profit. This
shows that the owner’s residual, the firm profit, is linear-quadratic in the contract:
π = (1 − b)b − na. Thus, the optimal contract for the owner, within this linear class
- the unique subgame-perfect equilibrium contract - is (a, b) = (0, 1/2), that is, zero
salary and the bonus rate b = 1/2. The unique subgame-perfect equilibrium effort is
thus xi = 0.5/n for all workers i.
    This equilibrium effort level can be contrasted with the effort level the workers
would like to commit to if they could, namely the level which maximizes the sum of
their utilities. It is easily verified that this maximum, under any given contract (a, b)
is obtained when every worker exerts effort xi = b. We call this the team optimum
effort under contract (a, b). At this team optimum, each worker thus exerts n times
more effort than in the absence of commitment possibilities.6 The discrepancy is due
to the equilibrium temptation to free-ride on each others’ work efforts.
   Remark 1: The simplest possible social norm is when each worker, ceteris
paribus, wants to exert neither less nor more effort than the others. In other words,
each worker experiences disutility from sticking out in terms of work effort. Such a
group norm can be easily incorporated in the present model as follows:
  6
     Interestingly, when the workers can commit to a common effort level, the optimum contract
is the same as without commitment power, since then we would have xi = b for all i, and thus
π = n (1 − b) b − na, implying a = 0 and b = 1/2.
                             Social norms and incentives in firms                                   5



                                        1
                              ui = w − x2 − v (x−i − xi )
                                        2 i
for some convex function v : R → R which has its minimum at zero. If this disutility
function v is differentiable, then v0 (0) = 0, and the first-order condition for worker i
becomes
                                      b
                                 xi = + v 0 (x−i − xi ) .
                                      n
Hence, a necessary and sufficient condition for symmetric (subgame) Nash equilib-
rium, where all workers exert the same effort x, is simply x = b/n + v 0 (0) = b/n, just
as in the absence of any social norm. A social norm of this particular form can thus
not explain other behaviors than those generated by the benchmark model based on
pure economic incentives.7 We therefore consider a slightly more complex variety of
group norms.
2.2. With a group norm. We now include the following group norm, or “work
ethic,” to each worker’s preferences: to exert the team optimum effort level, xi =
ˆ
x = b. In other words, under any given contract (a, b), each worker feels that he or
                                                       ˆ
she should ideally exert the same (high) effort level x = b - the effort that maximizes
all workers’ utility. Moreover, we assume that a worker’s embarrassment or disutility
of exerting less effort than this ideal is larger the closer other workers adhere to the
norm - that increases the peer pressure.
    More exactly, we assume that each worker’s preferences can be represented in the
following additively separable way:
                   1    1
           ui = w − x2 − [max(ˆ − xi , 0) − max(ˆ − x−i , 0)]2 g (ˆ − x−i ) ,
                     i        x                 x                 x
                   2    2
where the weight g (ˆ − x−i ) attached to one’s own downwards deviation is a contin-
                       x
uous function of the others’ downwards deviation. We assume that the more others
deviate, the less embarrassing one’s own deviation is. Hence, g : R → R+ is non-
increasing (by construction, it only matters how g behaves on R+ ). In other words,
g (ˆ − x−i ) is the intensity of the social norm, or the peer pressure, as felt by worker i
   x
when the others’ mean effort is x−i , and this pressure is higher the closer the others’
efforts are to the norm.8
   7
      Note, however, that if v is not differentiable at zero, then the social norm may have an effect.
For instance, if v (z) = α |z| for some α > 0, then any effort level x such that b/n − α ≤ x ≤ b/n + α
is an equilibrium level.
    8
      By contrast, Kandel and Lazear (1992) model workers’ utility as additive in wage earnings,
disutility of effort, and social pressure.
                            Social norms and incentives in firms                              6


   The earlier strategic independence of effort in the absence of the norm (and the
assumed constant returns to scale production) is now lost. A worker’s best choice of
effort here depends on what other workers do. Thus, the timing of effort decisions is
now relevant. In the following, we assume that workers decide simultaneously. More
exactly, we solve for subgame perfect equilibria in the game where the owner (princi-
pal) first chooses a contract whereupon all workers observe this and simultaneously
choose their efforts.
   Given any contract (a, b), worker i thus solves9

                                  b     1    1
                        max         xi − x2 − (b − xi )2 g (b − x−i ) .
                                          i
                        0≤xi ≤b   n     2    2
It is easily verified that the unique solution for each worker i is

                                             g(b − x−i ) + 1/n
                                    xi = b                     .
                                              g(b − x−i ) + 1

 Like in the benchmark model, the fixed salary a has no influence on effort, and hence
it is still optimal for the owner to set a = 0.
     We focus on symmetric Nash equilibria, that is, effort profiles in which all workers
exert the same effort x. Given any bonus rate b and team size n, the set of such Nash
equilibria in the subgame is characterized by the fixed-point equation x = G(x),
where G : R+ → [0, b] is the continuous and increasing function defined by

                                               g(b − x) + 1/n
                                   G(x) = b                   .                            (1)
                                                g(b − x) + 1
Each fixed point x is the individual effort level in a symmetric Nash equilibrium, and
vice versa.
    It is easily verified that for every b there exists at least one fixed point x, and that
no equilibrium effort is lower than in the absence of the social norm. Moreover, if the
peer-pressure function g is twice differentiable and concave, then so is G, and hence
multiplicity of equilibria is excluded in such cases. More generally:

      Proposition: There exists at least one fixed point x. If x is a fixed point,
      then b/n ≤ x < b. If g is twice differentiable with 2 [g0 ]2 = [1 + g] g 00 for
      all x ∈ (0, b), then there exists exactly one fixed point.
   9
     Constant terms have been dropped. As will be seen below, it is never optimal to exceed the
norm.
                              Social norms and incentives in firms                       7


    Proof: The first two claims follow from G being continuous with b/n < G (x) < b
for all x. ¡ The third claim follows from a straight-forward calculation: G00 (x) =
                                    ¢
b (1/n − 1) 2 [g 0 ]2 − [1 + g] g 00 / [1 + g]3 . End of proof.
    In case of uniqueness, the economist’s intuition that effort should be increasing
in economic incentives, here the bonus rate, is correct.10 In the case of multiple
equilibria, however, this intuition may be wrong, as we will show below.
    If there were no embarrassment of deviating from the social norm, i.e., if g(x) ≡ 0,
then the unique equilibrium effort would be x = b/n, just as in the benchmark
model. More generally, if the embarrassment were independent of others’ efforts,
g(b − x) ≡ θ ≥ 0, then the unique equilibrium effort,

                                                 θ + 1/n
                                            x=           b
                                                  θ+1
decreases in team size n but increases with the weight θ attached to the group norm,
from the norm-free equilibrium level x = b/n when θ = 0 towards the team optimum
x = b as θ tends to plus infinity.
   We finally note that the subgame equilibria associated with any given bonus rate
are Pareto ranked. For not only does an increase in effort, given a and b, yield a
higher profit to the owner, it also yields higher utility to all workers. To see this, note
that if they exert the same effort x < b, then every worker’s utility is simply
                                                    1
                                            u = bx − x2 ,
                                                    2
a concave function (a parabola turned upside down) with maximum at the team
optimum, b. Thus u increases with x for all x < b.

                                  3. Example
One class of differentiable weight functions g which have a simple parametrization
and meet the requirements (but, as will be seen shortly, are not concave), is given by
                            ¡                  ¢
                  g(z) = α/ β [max(z, 0)]2 + 1       for all z ∈ R,                (2)

where α, β ≥ 0. Here α ≥ 0 represents the weight placed on the social norm, as
compared with the economic incentive, and β represents the rate at which this weight
 10
      The implicit function theorem gives

                                dx   g 2 n + n + (1 − n)bng 0 g + g
                                   = 2                              .
                                db  g n + n + (1 − n)bng 0 g + 2ng
             dx
Hence, 0 ≤   db   ≤ 1.
                         Social norms and incentives in firms                        8


declines as others’ deviation z from the norm increases: the larger β is, the more
quickly does this weight decline with others’ average deviation, see Figure 1 below.




                  1


                 0.8


                 0.6


                 0.4


                 0.2


                  0     0.1   0.2      0.3   0.4   0.5   0.6   0.7    0.8

 Figure 1: The weight g (b − x), on the vertical axis, with x on the horizontal, for
     b = 0.7, α = 1, and β = 400 (solid), β = 200 (dash), and β = 800 (dot).


   Figure 2 shows the graph of the associated function G (for n = 10 and α = 30),
and with otherwise the same parameter values as in Figure 1. Hence, G is given by

                                    β [max(b − x, 0) + 1]2 /n + α
                       G (x) = b                                  .               (3)
                                     β [max(b − x, 0)]2 + 1 + α

This diagram shows that multiple equilibria may exist. For generic parameter values,
the number of equilibria is either 1 or 3. In case of three equilibria, only the high-
and low-effort equilibria are stable under adaptive dynamics; a small deviation from
the medium-effort equilibrium induces a movement towards either the high- or the
low-effort equilibrium level. We will therefore neglect the medium-effort equilibrium
in the subsequent discussion.
                                Social norms and incentives in firms                      9



                              0.8

                              0.7

                              0.6

                              0.5

                              0.4

                              0.3

                              0.2

                              0.1

                                0    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
                                                  x

   Figure 2: The fixed-point equation x = G(x), for b = 0.7, α = 30, n = 10, and
               β = 400 (solid), β = 200 (dash), and β = 900 (dot).


    Figure 3 plots the subgame-equilibrium correspondence - the set of subgame equi-
librium effort levels x (on the vertical axis) for each bonus rate b (on the horizontal
axis) in the unit interval, with the other parameters fixed at the same values as in
Figures 1 and 2. The graphs of the three equilibrium correspondences, one for each
value of β, are the three S-shaped curves in the diagram (for β = 800, 600, and 200,
respectively, moving from left to right). The steep straight line represents the team
optimum effort level, x = b, and the less steep straight line the equilibrium effort in
the absence of the social norm, x = b/n. The two hyperbolas are iso-profit curves.11
Tangency with such a hyperbola is thus a necessary (but in general not sufficient)
condition for optimum.
    The figure shows that the subgame equilibrium effort is unique for all bonus rates
when β = 400. By contrast, for β = 400, the equilibrium effort is unique as long as
the bonus rate is below a certain critical value, b1 , which is approximately 0.7. In
the interval (0, b1 ), the equilibrium effort is increasing in the bonus rate b. At the
critical value b1 there are two equilibria. Then follows an interval, (b1 , b2 ) with three
equilibria, with high, medium and low effort, respectively. At another critical bonus
rate, b2 , which is approximately 0.8 (still for β = 400), there are two equilibria, and
for all higher bonus rates the equilibrium effort is again unique.
  11
       Each iso-profit curve is of the form b = 1 − c/x, for c = π/n.
                        Social norms and incentives in firms                         10



                  1


                0.8


                0.6


                0.4


                0.2



                            0.2      0.4      0.6      0.8       1

Figure 3: Subgame equilibrium effort levels x (vertical axis) for different bonus rate
                               b (horizontal axis).


    As can be seen in the figure, the first-best bonus rate, that is the rate which maxi-
mizes profits across the whole equilibrium manifold (for β = 400, we have b∗ ≈ 0.73),
may entail multiple equilibria. Thus, a principal who chooses his bonus rate in this
way, in expectation of the corresponding high-effort equilibrium, might face a dis-
appointing surprise. If workers instead coordinate on the low-effort equilibrium at
that bonus rate, his profits will be considerably lower than anticipated. By contrast,
the owner may ensure a higher equilibrium profit than in this worst-case equilib-
rium scenario under the first-best bonus rate, by instead choosing a higher bonus
rate (above the second critical bonus rate, b2 ≈ 0.8, when β = 400), such that the
associated subgame equilibrium is unique. This defines the second-best bonus rate,
the rate which maximizes profits across the range of bonus rates which induce unique
subgame equilibria.
    Given any bonus rate for which there exists multiple subgame equilibria, which
of these is more likely to prevail? One potential avenue for such an investigation is
offered by evolutionary game theory: the interaction between the workers constitutes
                          Social norms and incentives in firms                            11


a coordination game, and the tools developed by Kandori-Mailath-Rob (1993), Young
(1993), and Benaim and Weibull (2001), among others, could be applied to a discrete-
choice version of the subgame. However, such an analysis falls outside the scope of
the present study. We note, though, that the high-effort equilibrium not only gives
the highest profit but, as noted above, also the highest utility to all workers. The
(symmetric) equilibrium with the highest equilibrium thus Pareto dominates the other
equilibria. Hence, bot (product) market selection and worker mobility speak against
the other equilibria: a lower profit makes the firm’s survival harder, and a lower utility
to the workers makes them less willing to keep their jobs. Indeed, if workers in the
relevant labor market have a reservation utility above that in a low-effort equilibrium
but below that in a high-effort equilibrium, and if transaction costs are low, then
workers will leave a firm which is trapped in the low-effort equilibrium.12
    We also note that the so-called crowding effect of economic incentives, as dis-
cussed by Frey (1997), do not always come to play in the present model. Frey argues
that increased economic incentives may “crowd out intrinsic motivation” and, thus,
ultimately reduce efforts. In the example in Figure 3 this does not happen. Suppose
the bonus rate is such that there are multiple equilibria and workers coordinate on
the high-effort equilibrium. Now suppose the bonus rate is increased. As the rate is
gradually increased, it seems reasonable to assume that workers will just marginally
adjust their effort. The way the fold of the equilibrium correspondence is turned in
this example, there is no sudden fall in effort as the bonus rate is increased. Other ex-
amples can be constructed where a small rise in the bonus rate results in a sudden and
large drop in effort. For an example, see Figure 4 below, where g (z) = α exp (−βz 2 ),
for α = 6 and β = 25.
  12
    It would be an interesting game-theoretic exercise to study whether risk-dominance works
against the high-effort equilibrium in such a situation.
                        Social norms and incentives in firms                        12



                  1


                0.8


                0.6


                0.4


                0.2



                           0.2       0.4       0.6       0.8     1

   Figure 4: The equilibrium correspondence, for an exponential peer-pressure
                                   function.

                         4. Variations and extensions
In this section we briefly study various alternatives to and extensions of our model,
in order to illuminate the robustness of the above qualitative results, and to see how
they can be generalized.

4.1. Decreasing returns to scale. How do our results depend on the linearity
of the production function? We here briefly consider the case of decreasing returns.
                                            P
More precisely, let y = f (L), where L = j xj and f : R+ → R+ is twice differen-
tiable, with f 0 > 0, f 00 < 0. The associated team optimum effort level is implicitly
defined by the equation bf 0 (nˆ) = x, and the fixed-point equation that determines
                                x     ˆ
(subgame) equilibrium efforts becomes x = G (x), where

                                    xg (ˆ − x) + bf 0 (nx)/n
                                    ˆ x
                          G (x) =                            .
                                         g (ˆ − x) + 1
                                            x

This function is identical with the function on the right-hand side of equation (1) in
the special case of constant returns to scale. We also note that the function G need
                          Social norms and incentives in firms                          13


no longer be increasing. In particular, if the production function f meets the usual
Inada condition that f 0 (x) → +∞ as x goes to zero, then also G (x) → +∞ as x
                                                                                     ˆ
goes to zero. Despite this, at least one fixed point still exists in the interval (0, x),
since G is continuous, with
                                         g (0) + 1/n
                                  x
                               G (ˆ) =               ˆ ˆ
                                                     x<x.
                                          g (0) + 1
4.2. Leontieff production technology. So far, all workers’ efforts have been
perfect substitutes. Consider now the extreme opposite case when individual efforts
are perfect complements. Hence, suppose y = minj xj . The team optimum level is
now the common individual effort x which maximizes bx − nx2 /2. Thus, x = b/n. As
                                                                            ˆ
will be seen below, in the absence of the group norm, any effort level in the interval
[0, b/n] is consistent with Nash equilibrium, and in the presence of the group norm,
the set of equilibrium effort levels is a subinterval of the form [c, b/n], where c > 0 is
larger the more important the social norm is in comparison with economic incentives.
     To see this, assume that all workers j 6= i choose the same effort level x. Then
the utility to worker i depends as follows on i’s own effort:
                      bxi /n − 1 x2 − 1 (b/n − xi )2 g (b/n − x) if xi ≤ x
                               2 i    2
               ui =                                                        .
                      bx/n − 1 x2 − 1 (b/n − xi )2 g (b/n − x) if xi > x
                               2 i    2

Hence, as a function of his own effort xi , his utility is continuous, but has a kink
precisely at the others’ effort level x. It is easily verified that a necessary and sufficient
condition for symmetric Nash equilibrium, where all workers choose the same effort
x, is given by the following double inequality:

                                 g (b/n − x) b      b
                                                 ≤x≤ .                                (4)
                               1 + g (b/n − x) n    n
    We note, in particular, that in the absence of the group norm (that is, when g is
identically equal to zero), this inequality is met by every effort level x in the whole
interval [0, b/n], as claimed above. We also note that, as the importance of the social
norm increases, the interval of Nash equilibrium effort levels is a shrinking interval,
which has the team optimum effort b/n as its right end point, and the left end of
which converges to this right end point. More precisely, if g is positive, and all its
values are multiplied by the same positive constant λ, then the interval’s left end
point increases with λ and converges to b/n as λ → ∞.
    In sum: the effect of a social norm in the case when all workers’ efforts are perfect
complements is to reduce the continuum set of equilibrium effort levels in the direction
towards the team optimum effort.
                         Social norms and incentives in firms                         14


4.3. Partnerships. A partnership, that is, a firm owned by workers who split
the profit in equal shares, is equivalent with setting b = 1 in our model. The team
                                     ˆ
optimum effort level thus becomes x = 1. Using the weight function g from the
example, the fixed-point equation that determines (symmetric) equilibrium effort
levels becomes x = G (x), where

                                         g(1 − x) + 1/n
                               G (x) =
                                          g(1 − x) + 1

    This fixed-point equation has qualitatively the same properties as in the basic
model. In particular, for certain peer-pressure functions there exist multiple equilib-
ria, and these may be Pareto ranked according to their effort levels.

                                    5. Discussion
This paper analyzes optimal incentive schemes for teams, within a class of linear
contracts, in the presence of a group work-norm. In the simple model we investigate,
there is a moral hazard problem, due to the principal choosing a bonus rate below
one, and there is a free-rider problem among the team members. The work norm
mitigates the free-rider problem by causing those whose effort falls short of the team
optimum to suffer a utility loss. The examples show that the optimal bonus rate
can be higher or lower than in the absence of the social norm. The intuition for
this finding is straightforward: There are two countervailing effects of the social
norm on the optimal bonus. First, an increase in the bonus rate not only increases
the economic incentive to each worker (∆xi = ∆b/n), but also indirectly increases
the “social incentive”. If others work harder (because of their increased economic
incentives), then a worker wants to increase his effort too in order to reduce his
increased social embarrassment (due to the raised work norm). Second, the norm
increases the effort level directly, making the bonus less important for eliciting effort.
    The multiplicity of equilibria associated with a given bonus rate suggest a dis-
tinction between first- and second best optimal bonus rates, where first-best is the
maximum across the whole subgame equilibrium manifold and second-best is the
maximum across those part of the correspondence where there is a unique subgame
equilibrium effort level. The multiplicity of equilibria also suggests a dynamic per-
spective. For example, suppose the firm pays a bonus rate at which multiple equilibria
co-exist, but a low-effort equilibrium is realized. To move away from this inefficient
equilibrium, the firm owner may temporarily increase the bonus to a level where the
equilibrium is unique (see Figures 3 and 4). Workers’ efforts may then jump up to
this unique equilibrium level. Assuming that workers adapt gradually to gradual
changes in the bonus b, along the current branch of the equilibrium correspondence,
                             Social norms and incentives in firms                                     15


the owner may afterwards gradually reduce the bonus, even to its original value, but
now at the efficient, high-effort, equilibrium.13
    In a future project we plan to study team production game protocols in a series
of laboratory experiments.14 While we conjecture that this will confirm the relevance
of social norms, we also expect that institutional detail will influence the evolution
and strength of such norms.15 Another avenue for future work is to use tools from
evolutionary game theory to (a) analyze the relative stability of alternative subgame
equilibria in case of multiplicity, (b) endogenize the social norm for work effort, which
here was exogenously set at the team optimum level.

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     See sections II and VII in Lindbeck et al. (1999) for a discussion of similar equilibrium dynamics.
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                              a
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  15
     See, for example, recent work by Bohnet, Frey, and Huck (2001) who study the evolution of
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