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Social norms and incentives in ļ¬rms Steffen Huck, Dorothea Kubler and Jorgen Weibullā ¨ ¨ 4 May 2002. Revised version of IUI WP 565, 2001. Abstract. This paper studies the interplay between economic incentives and social work norms in ļ¬rms. 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 ļ¬rms 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 aļ¬ect the incentive structure within ļ¬rms.2 This note outlines a new attempt in this direction. For a ļ¬rm owner, social norms concerning work are important because they can aļ¬ect proļ¬ts. For example, norms may inļ¬uence how much eļ¬ort workers put into projects where only joint output is observable. Social norms, emanating from the ratchet eļ¬ect, may also keep workers from working hard under relative-performance ā 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), Moļ¬tt (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 ļ¬rms, but rather deals with the question whether the degree of trust between agents inļ¬uences the optimal ownership structure. 1 Social norms and incentives in ļ¬rms 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 eļ¬ort level, but also on the eļ¬ort 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 agentās eļ¬ort level is unobserved by the principal, but total output can be observed and veriļ¬ed. The principal chooses among linear incentive schemes in order to maxi- mize proļ¬ts, and the agents in the team simultaneously choose their eļ¬orts thereafter. We study the eļ¬ect of a social norm concerning work eļ¬ort 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) ļ¬nd substantial shirking diļ¬erentials between branches of an Italian ļ¬rm, despite identical monetary incentives governing the employeesā eļ¬orts in these branches. They identify group- interaction eļ¬ects as a key explanatory variable that allows for multiple equilibria. However, they do not analyze the interplay of these group-interaction eļ¬ects with economic incentives. Encinosa, Gaynor, and Rebitzer (1997) ļ¬nd 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 agentsā 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 eļ¬orts and low social pressure can coexist, under the same incentive scheme, with equilibria with high eļ¬orts and high social pressure. This multiplicity is relevant for a principal who strives to ļ¬nd a proļ¬t-maximizing incentive scheme. For example, a ļ¬rm trapped in a low-eļ¬ort equilibrium may ājump upā to a high-eļ¬ort 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 beneļ¬cial for ļ¬rm owners to temporarily change the economic incentive scheme, or in some other way temporarily inspire the workers to high eļ¬orts (even at a high temporary cost) in order to lead the team to the basin of attraction of a high-eļ¬ort equilibrium. Such ājump startingā may lead the team away from a low-eļ¬ort equilibrium. Afterwards, 3 For a pioneering model of team production, see Holmstrom (1982). Social norms and incentives in ļ¬rms 3 the temporary stimulus may be withdrawn while the workersā eļ¬orts 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 eļ¬ort, so also the workers prefer the high-eļ¬ort (and thus high proļ¬t) 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 modelās general qualitative properties. These general observations are illustrated in some detail in an example in section 3. Section 4 discusses brieļ¬y some variations and generalizations of the model assumptions. Section 5 concludes. 2. The model We consider team production with a proļ¬t maximizing owner (the principal) as resid- ual claimant. There are n > 1 identical workers (agents). Each worker i exerts some eļ¬ort xi ā„ 0. Let xāi denote the average eļ¬ort exerted by all other workers, P xāi = j6=i xj / (n ā 1). The production technology is linear: output y equals the P sum of all workersā eļ¬orts, y = n xi . The principal can only observe aggregate i=1 output y, not individual eļ¬orts. The workers are also assumed to observe aggregate eļ¬ort, from which each worker i can deduce the average eļ¬ort xāi of the others (as- suming knowledge of the production technology and his or her own eļ¬ort). 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 aļ¬ne function of the ļ¬rmās output, w = a + by/n . where the owner chooses the ļ¬xed salary a and the bonus rate b. We require a to be nonnegative, an assumption which can be justiļ¬ed by wealth constraints.5 Therefore, the proļ¬t-maximizing owner will optimally choose b in the open unit interval, 0 < b < 1. Assuming that the ļ¬rm is a price taker in its product market, and normalizing the market price to unity, the ļ¬rmās proļ¬t ā the residual left to the owner ā is thus Ļ = (1 ā b)y ā 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 ļ¬rm to each of the workers. In the absence of social norms, such an arrangement would make the ļ¬rst-best solution achievable. However, such schemes are vulnerable to collusion and sabotage. Social norms and incentives in ļ¬rms 4 2.1. Without a group norm. We ļ¬rst analyze the benchmark case when social norms are absent or have no inļ¬uence on behavior. A workerās utility then only depends on his or her wage earning and eļ¬ort. We assume throughout this study that the workers are identical and have linear-quadratic utility in consumption and work-eļ¬ort: 1 b 1 ui = w ā x2 = a + [(n ā 1) xāi + xi ] ā x2 . i 2 n 2 i From this it is immediate that workersā decisions concerning eļ¬ort are strategically independent. Regardless of whether workers decide simultaneously or sequentially, each worker solves b 1 max xi ā x2 . xi ā„0 n 2 i Consequently, the unique equilibrium eļ¬ort level, given any contract (a, b), are xi = b/n for all workers i. Viewing workersā eļ¬ort choices as a subgame following upon the ownerās choice of contract, we solve the full game for subgame perfect equilibrium by inserting the above (subgame equilibrium) eļ¬ort levels into the expression for the ļ¬rmās proļ¬t. This shows that the ownerās residual, the ļ¬rm proļ¬t, is linear-quadratic in the contract: Ļ = (1 ā b)b ā 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 eļ¬ort is thus xi = 0.5/n for all workers i. This equilibrium eļ¬ort level can be contrasted with the eļ¬ort level the workers would like to commit to if they could, namely the level which maximizes the sum of their utilities. It is easily veriļ¬ed that this maximum, under any given contract (a, b) is obtained when every worker exerts eļ¬ort xi = b. We call this the team optimum eļ¬ort under contract (a, b). At this team optimum, each worker thus exerts n times more eļ¬ort than in the absence of commitment possibilities.6 The discrepancy is due to the equilibrium temptation to free-ride on each othersā work eļ¬orts. Remark 1: The simplest possible social norm is when each worker, ceteris paribus, wants to exert neither less nor more eļ¬ort than the others. In other words, each worker experiences disutility from sticking out in terms of work eļ¬ort. Such a group norm can be easily incorporated in the present model as follows: 6 Interestingly, when the workers can commit to a common eļ¬ort level, the optimum contract is the same as without commitment power, since then we would have xi = b for all i, and thus Ļ = n (1 ā b) b ā na, implying a = 0 and b = 1/2. Social norms and incentives in ļ¬rms 5 1 ui = w ā x2 ā v (xāi ā xi ) 2 i for some convex function v : R ā R which has its minimum at zero. If this disutility function v is diļ¬erentiable, then v0 (0) = 0, and the ļ¬rst-order condition for worker i becomes b xi = + v 0 (xāi ā xi ) . n Hence, a necessary and suļ¬cient condition for symmetric (subgame) Nash equilib- rium, where all workers exert the same eļ¬ort 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 āwork ethic,ā to each workerās preferences: to exert the team optimum eļ¬ort level, xi = Ė x = b. In other words, under any given contract (a, b), each worker feels that he or Ė she should ideally exert the same (high) eļ¬ort level x = b - the eļ¬ort that maximizes all workersā utility. Moreover, we assume that a workerās embarrassment or disutility of exerting less eļ¬ort than this ideal is larger the closer other workers adhere to the norm - that increases the peer pressure. More exactly, we assume that each workerās preferences can be represented in the following additively separable way: 1 1 ui = w ā x2 ā [max(Ė ā xi , 0) ā max(Ė ā xāi , 0)]2 g (Ė ā xāi ) , i x x x 2 2 where the weight g (Ė ā xāi ) attached to oneās own downwards deviation is a contin- x uous function of the othersā downwards deviation. We assume that the more others deviate, the less embarrassing oneās own deviation is. Hence, g : R ā R+ is non- increasing (by construction, it only matters how g behaves on R+ ). In other words, g (Ė ā xāi ) is the intensity of the social norm, or the peer pressure, as felt by worker i x when the othersā mean eļ¬ort is xāi , and this pressure is higher the closer the othersā eļ¬orts are to the norm.8 7 Note, however, that if v is not diļ¬erentiable at zero, then the social norm may have an eļ¬ect. For instance, if v (z) = Ī± |z| for some Ī± > 0, then any eļ¬ort level x such that b/n ā Ī± ā¤ x ā¤ b/n + Ī± is an equilibrium level. 8 By contrast, Kandel and Lazear (1992) model workersā utility as additive in wage earnings, disutility of eļ¬ort, and social pressure. Social norms and incentives in ļ¬rms 6 The earlier strategic independence of eļ¬ort in the absence of the norm (and the assumed constant returns to scale production) is now lost. A workerās best choice of eļ¬ort here depends on what other workers do. Thus, the timing of eļ¬ort 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) ļ¬rst chooses a contract whereupon all workers observe this and simultaneously choose their eļ¬orts. Given any contract (a, b), worker i thus solves9 b 1 1 max xi ā x2 ā (b ā xi )2 g (b ā xāi ) . i 0ā¤xi ā¤b n 2 2 It is easily veriļ¬ed that the unique solution for each worker i is g(b ā xāi ) + 1/n xi = b . g(b ā xāi ) + 1 Like in the benchmark model, the ļ¬xed salary a has no inļ¬uence on eļ¬ort, and hence it is still optimal for the owner to set a = 0. We focus on symmetric Nash equilibria, that is, eļ¬ort proļ¬les in which all workers exert the same eļ¬ort x. Given any bonus rate b and team size n, the set of such Nash equilibria in the subgame is characterized by the ļ¬xed-point equation x = G(x), where G : R+ ā [0, b] is the continuous and increasing function deļ¬ned by g(b ā x) + 1/n G(x) = b . (1) g(b ā x) + 1 Each ļ¬xed point x is the individual eļ¬ort level in a symmetric Nash equilibrium, and vice versa. It is easily veriļ¬ed that for every b there exists at least one ļ¬xed point x, and that no equilibrium eļ¬ort is lower than in the absence of the social norm. Moreover, if the peer-pressure function g is twice diļ¬erentiable and concave, then so is G, and hence multiplicity of equilibria is excluded in such cases. More generally: Proposition: There exists at least one ļ¬xed point x. If x is a ļ¬xed point, then b/n ā¤ x < b. If g is twice diļ¬erentiable with 2 [g0 ]2 = [1 + g] g 00 for all x ā (0, b), then there exists exactly one ļ¬xed 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 ļ¬rms 7 Proof: The ļ¬rst 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 ā 1) 2 [g 0 ]2 ā [1 + g] g 00 / [1 + g]3 . End of proof. In case of uniqueness, the economistās intuition that eļ¬ort 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) ā” 0, then the unique equilibrium eļ¬ort would be x = b/n, just as in the benchmark model. More generally, if the embarrassment were independent of othersā eļ¬orts, g(b ā x) ā” Īø ā„ 0, then the unique equilibrium eļ¬ort, Īø + 1/n x= b Īø+1 decreases in team size n but increases with the weight Īø attached to the group norm, from the norm-free equilibrium level x = b/n when Īø = 0 towards the team optimum x = b as Īø tends to plus inļ¬nity. We ļ¬nally note that the subgame equilibria associated with any given bonus rate are Pareto ranked. For not only does an increase in eļ¬ort, given a and b, yield a higher proļ¬t to the owner, it also yields higher utility to all workers. To see this, note that if they exert the same eļ¬ort x < b, then every workerās utility is simply 1 u = bx ā 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 diļ¬erentiable 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) = Ī±/ Ī² [max(z, 0)]2 + 1 for all z ā R, (2) where Ī±, Ī² ā„ 0. Here Ī± ā„ 0 represents the weight placed on the social norm, as compared with the economic incentive, and Ī² represents the rate at which this weight 10 The implicit function theorem gives dx g 2 n + n + (1 ā n)bng 0 g + g = 2 . db g n + n + (1 ā n)bng 0 g + 2ng dx Hence, 0 ā¤ db ā¤ 1. Social norms and incentives in ļ¬rms 8 declines as othersā deviation z from the norm increases: the larger Ī² is, the more quickly does this weight decline with othersā 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 ā x), on the vertical axis, with x on the horizontal, for b = 0.7, Ī± = 1, and Ī² = 400 (solid), Ī² = 200 (dash), and Ī² = 800 (dot). Figure 2 shows the graph of the associated function G (for n = 10 and Ī± = 30), and with otherwise the same parameter values as in Figure 1. Hence, G is given by Ī² [max(b ā x, 0) + 1]2 /n + Ī± G (x) = b . (3) Ī² [max(b ā x, 0)]2 + 1 + Ī± 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-eļ¬ort equilibria are stable under adaptive dynamics; a small deviation from the medium-eļ¬ort equilibrium induces a movement towards either the high- or the low-eļ¬ort equilibrium level. We will therefore neglect the medium-eļ¬ort equilibrium in the subsequent discussion. Social norms and incentives in ļ¬rms 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 ļ¬xed-point equation x = G(x), for b = 0.7, Ī± = 30, n = 10, and Ī² = 400 (solid), Ī² = 200 (dash), and Ī² = 900 (dot). Figure 3 plots the subgame-equilibrium correspondence - the set of subgame equi- librium eļ¬ort levels x (on the vertical axis) for each bonus rate b (on the horizontal axis) in the unit interval, with the other parameters ļ¬xed at the same values as in Figures 1 and 2. The graphs of the three equilibrium correspondences, one for each value of Ī², are the three S-shaped curves in the diagram (for Ī² = 800, 600, and 200, respectively, moving from left to right). The steep straight line represents the team optimum eļ¬ort level, x = b, and the less steep straight line the equilibrium eļ¬ort in the absence of the social norm, x = b/n. The two hyperbolas are iso-proļ¬t curves.11 Tangency with such a hyperbola is thus a necessary (but in general not suļ¬cient) condition for optimum. The ļ¬gure shows that the subgame equilibrium eļ¬ort is unique for all bonus rates when Ī² = 400. By contrast, for Ī² = 400, the equilibrium eļ¬ort 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 eļ¬ort 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 eļ¬ort, respectively. At another critical bonus rate, b2 , which is approximately 0.8 (still for Ī² = 400), there are two equilibria, and for all higher bonus rates the equilibrium eļ¬ort is again unique. 11 Each iso-proļ¬t curve is of the form b = 1 ā c/x, for c = Ļ/n. Social norms and incentives in ļ¬rms 10 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 Figure 3: Subgame equilibrium eļ¬ort levels x (vertical axis) for diļ¬erent bonus rate b (horizontal axis). As can be seen in the ļ¬gure, the ļ¬rst-best bonus rate, that is the rate which maxi- mizes proļ¬ts across the whole equilibrium manifold (for Ī² = 400, we have bā ā 0.73), may entail multiple equilibria. Thus, a principal who chooses his bonus rate in this way, in expectation of the corresponding high-eļ¬ort equilibrium, might face a dis- appointing surprise. If workers instead coordinate on the low-eļ¬ort equilibrium at that bonus rate, his proļ¬ts will be considerably lower than anticipated. By contrast, the owner may ensure a higher equilibrium proļ¬t than in this worst-case equilib- rium scenario under the ļ¬rst-best bonus rate, by instead choosing a higher bonus rate (above the second critical bonus rate, b2 ā 0.8, when Ī² = 400), such that the associated subgame equilibrium is unique. This deļ¬nes the second-best bonus rate, the rate which maximizes proļ¬ts 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 oļ¬ered by evolutionary game theory: the interaction between the workers constitutes Social norms and incentives in ļ¬rms 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-eļ¬ort equilibrium not only gives the highest proļ¬t 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 proļ¬t makes the ļ¬rmās 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-eļ¬ort equilibrium but below that in a high-eļ¬ort equilibrium, and if transaction costs are low, then workers will leave a ļ¬rm which is trapped in the low-eļ¬ort equilibrium.12 We also note that the so-called crowding eļ¬ect 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 ācrowd out intrinsic motivationā and, thus, ultimately reduce eļ¬orts. 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-eļ¬ort 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 eļ¬ort. The way the fold of the equilibrium correspondence is turned in this example, there is no sudden fall in eļ¬ort 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 eļ¬ort. For an example, see Figure 4 below, where g (z) = Ī± exp (āĪ²z 2 ), for Ī± = 6 and Ī² = 25. 12 It would be an interesting game-theoretic exercise to study whether risk-dominance works against the high-eļ¬ort equilibrium in such a situation. Social norms and incentives in ļ¬rms 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 brieļ¬y 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 brieļ¬y consider the case of decreasing returns. P More precisely, let y = f (L), where L = j xj and f : R+ ā R+ is twice diļ¬eren- tiable, with f 0 > 0, f 00 < 0. The associated team optimum eļ¬ort level is implicitly deļ¬ned by the equation bf 0 (nĖ) = x, and the ļ¬xed-point equation that determines x Ė (subgame) equilibrium eļ¬orts becomes x = G (x), where xg (Ė ā x) + bf 0 (nx)/n Ė x G (x) = . g (Ė ā 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 ļ¬rms 13 no longer be increasing. In particular, if the production function f meets the usual Inada condition that f 0 (x) ā +ā as x goes to zero, then also G (x) ā +ā as x Ė goes to zero. Despite this, at least one ļ¬xed point still exists in the interval (0, x), since G is continuous, with g (0) + 1/n x G (Ė) = Ė Ė x<x. g (0) + 1 4.2. Leontieļ¬ production technology. So far, all workersā eļ¬orts have been perfect substitutes. Consider now the extreme opposite case when individual eļ¬orts are perfect complements. Hence, suppose y = minj xj . The team optimum level is now the common individual eļ¬ort x which maximizes bx ā nx2 /2. Thus, x = b/n. As Ė will be seen below, in the absence of the group norm, any eļ¬ort level in the interval [0, b/n] is consistent with Nash equilibrium, and in the presence of the group norm, the set of equilibrium eļ¬ort 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 eļ¬ort level x. Then the utility to worker i depends as follows on iās own eļ¬ort: bxi /n ā 1 x2 ā 1 (b/n ā xi )2 g (b/n ā x) if xi ā¤ x 2 i 2 ui = . bx/n ā 1 x2 ā 1 (b/n ā xi )2 g (b/n ā x) if xi > x 2 i 2 Hence, as a function of his own eļ¬ort xi , his utility is continuous, but has a kink precisely at the othersā eļ¬ort level x. It is easily veriļ¬ed that a necessary and suļ¬cient condition for symmetric Nash equilibrium, where all workers choose the same eļ¬ort x, is given by the following double inequality: g (b/n ā x) b b ā¤xā¤ . (4) 1 + g (b/n ā 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 eļ¬ort 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 eļ¬ort levels is a shrinking interval, which has the team optimum eļ¬ort 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 Ī», then the intervalās left end point increases with Ī» and converges to b/n as Ī» ā ā. In sum: the eļ¬ect of a social norm in the case when all workersā eļ¬orts are perfect complements is to reduce the continuum set of equilibrium eļ¬ort levels in the direction towards the team optimum eļ¬ort. Social norms and incentives in ļ¬rms 14 4.3. Partnerships. A partnership, that is, a ļ¬rm owned by workers who split the proļ¬t in equal shares, is equivalent with setting b = 1 in our model. The team Ė optimum eļ¬ort level thus becomes x = 1. Using the weight function g from the example, the ļ¬xed-point equation that determines (symmetric) equilibrium eļ¬ort levels becomes x = G (x), where g(1 ā x) + 1/n G (x) = g(1 ā x) + 1 This ļ¬xed-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 eļ¬ort 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 eļ¬ort falls short of the team optimum to suļ¬er 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 ļ¬nding is straightforward: There are two countervailing eļ¬ects of the social norm on the optimal bonus. First, an increase in the bonus rate not only increases the economic incentive to each worker (āxi = āb/n), but also indirectly increases the āsocial incentiveā. If others work harder (because of their increased economic incentives), then a worker wants to increase his eļ¬ort too in order to reduce his increased social embarrassment (due to the raised work norm). Second, the norm increases the eļ¬ort level directly, making the bonus less important for eliciting eļ¬ort. The multiplicity of equilibria associated with a given bonus rate suggest a dis- tinction between ļ¬rst- and second best optimal bonus rates, where ļ¬rst-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 eļ¬ort level. The multiplicity of equilibria also suggests a dynamic per- spective. For example, suppose the ļ¬rm pays a bonus rate at which multiple equilibria co-exist, but a low-eļ¬ort equilibrium is realized. To move away from this ineļ¬cient equilibrium, the ļ¬rm owner may temporarily increase the bonus to a level where the equilibrium is unique (see Figures 3 and 4). Workersā eļ¬orts 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 ļ¬rms 15 the owner may afterwards gradually reduce the bonus, even to its original value, but now at the eļ¬cient, high-eļ¬ort, 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 conļ¬rm the relevance of social norms, we also expect that institutional detail will inļ¬uence 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 eļ¬ort, which here was exogenously set at the team optimum level. References [1] Akerlof G. (1980). 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