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. References  Akerlof G. (1980). 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