Productivity effects of innovation, stress and social relations

Journal of Economic Behavior & Organization 79 (2011) 165–182

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Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/jebo

Productivity effects of innovation, stress and social relations夽 Robin Cowan a,b,∗ , Bulat Sanditov a,c , Rifka Weehuizen a a b c

UNU-MERIT, Maastricht University, Netherlands BETA, Université Louis Pasteur, Strasbourg, France CESPRI, Università Bocconi, Milan, Italy

a r t i c l e

i n f o

Article history: Received 26 August 2008 Received in revised form 17 June 2010 Accepted 31 January 2011 Available online 11 March 2011 JEL classification: O31 O33

a b s t r a c t Innovation is a source of increasing productivity, but also of stress. Psychological research shows that individual productivity increases and then decreases as stress levels increase. Agents’ stress levels are determined by their own coping ability and by positive and negative spillovers to their social contacts. We model stress and inter-agent dynamics, identifying the relationships between innovation, stress and productivity. We characterize conditions under which multiple equilibria in stress levels and growth rates exist; and under which the dynamics exhibit hysteresis. High rates of innovation can result in high stress equilibrium and have a negative effect on economic growth. © 2011 Elsevier B.V. All rights reserved.

Keywords: Innovation Stress Social relationships

1. Introduction It is generally assumed that innovation leads to higher productivity and more economic growth, and that economic growth leads to increased well-being. These assumptions form the implicit and explicit basis for the justification of a vast array of policy aimed at increasing innovation to increase competitiveness and welfare. There are, however, several observations that suggest that this conclusion may be hasty. Innovation involves change and novelty, and these are associated with stress. But a vast body of research in psychology shows that above some level, stress is associated with deteriorating performance, and mental and physical health problems; and this in turn is associated with reduced productivity. This suggests that blindly pursuing higher innovation rates may eventually back-fire: the increased amount of change that workers have to deal with will eventually reduce their productivity, and so defeat the purpose of the innovation. In addition, stress spills over to other actors and has a negative effect on social relations, which in turn can lead to yet more stress. Policy aimed at more innovation and more creative destruction in order to strengthen economic growth, may thus have unintended effects not only on social well-being but also on economic growth itself. Estimates of the cost of work related stress are as high as four percent of GDP (Hoel et al., 2001), and problems associated with occupational stress have been a growing concern for policy makers throughout the industrialized world (see for example European Commission, 1999). It has largely been ignored by economists though. A rare exception is Greiner (2008) who studies a model of optimal consumption and labour supply in the presence of stress. In this model stress is accumulated

夽 We are grateful to participants of Behavioural and Experimental Economics (BEE) seminar at Maastricht University and “Policies for Happiness” conference (Siena, June 2007) and two anonymous referees for their useful comments and suggestions. ∗ Corresponding author. E-mail addresses: [email protected] (R. Cowan), [email protected] (B. Sanditov), [email protected] (R. Weehuizen). 0167-2681/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jebo.2011.01.028

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over working hours and has a non-monotonic effect on the agent’s well being. Introduction of stress leads to non-trivial dynamics, and in particular, cyclical behaviour of labour and consumption. The effects of social relationships on stress, though mentioned, are not included in the model. In the current paper, we model a group of agents in an innovative environment for whom innovation is a source of stress. The agents, however, are involved in social relationships which include both stress spillover, and stress buffering. We show that variables moderating stress levels through that relationship affect the relationship between innovation and productivity. In addition, we show the possibility of multiple equilibria in stress levels and that the dynamics exhibit hysteresis. The paper is organized as follows. First we identify and discuss the different variables at stake, and their relationships (Section 2). Formalizing these relationships, we derive results relating the interaction effects on overall stress levels with the effect of innovation on productivity growth (Section 3). In Section 4 we discuss how different parameters affect the behaviour of the model. Section 5 concludes. 2. Innovation, stress, and social relationships The essence of the model is that innovation creates stress, and that stress has a non-monotonic effect on productivity. For any agent, stress can be reduced both by that agent invoking (autonomous) coping mechanisms, and through the buffering effects of his or her social contacts. But at the same time stress can spill over from one agent to another, again through social contacts. In this section we review the literature, largely from psychology, that supports these claims. 2.1. Innovation and stress Innovation by defnition involves change and novelty, and there is a large body of empirical evidence showing that this induces stress. A standard model in the stress literature is the demand-control model of Karasek (1979) in which the level of stress experienced by an agent is determined by discrepancy between (perceived) demands and (perceived) control. There are several ways in which innovation tends to lead to stress, through increasing the mental demands workers face. Firstly, innovations always implies a degree of change and often uncertainty, which tends to be experienced as a form of stress, needed to trigger the individual to be alert and to adjust to the changing circumstances (e.g. Selye, 1956; Monat et al., 1972; Rabkin and Struening, 1976; Rafferty and Griffin, 2006). In general it is mainly the valence of the change, the perception of whether it is a desirable change or not, which determines to what extent change leads to stress (Mirowsky and Ross, 2003). Uncertainty resulting from innovation is generally experienced as stressful, though personal and contextual factors are obviously important moderators for the extent to which this is the case (e.g. Evers et al., 2002; Vieitez et al., 2001). Secondly, innovation implies novelty and novelty reduces the extent to which an agent can rely on routines. Routines economise on scarce information processing and decision-making capacity of agents (Simon, 1947, 1955, 1977). They coordinate behaviour and bring predictability and implicit agreement on how to act (Nelson and Winter, 1982). This smoothes interaction, enabling efficient and effective cooperation (March and Olsen, 1989). When circumstances change, existing routines will be less effective or possibly even counterproductive, and as a consequence agents face more demands in terms of time and (mental) resources to perform what had previously been routine activities (Alterman and Zito-Wolf, 1993). Thirdly, innovation often replaces repetitive, routine (manual and mental) tasks with mechanization and automatization. If routine tasks are successfully substituted by technology, the ratio of non-routine to routine tasks in a job increases. By definition non-routine tasks involve greater mental resources (demands) than do routine ones. In combination with complementary changes in management practices this makes work on average more demanding, both mentally and emotionally (Hochschild, 1983; Morris and Feldman, 1996; Glomb et al., 2004), a phenomenon that is called the ‘intensification of work’ (Green, 2001; Green and McIntosh, 2001). Thus, while it is not possible simply to equate innovation with change, and change with stress, there is sufficient rationale and evidence to assume that innovation often leads to lead to a degree of stress, through creation of uncertainty, the increased workload as a result of the need to adjust and build new routines, and the intensification of work after innovation. Obviously, some innovations are designed explicitly to improve working life and to reduce stress, and are aimed at streamlining operations and reducing excess tasks. Any innovation of this sort will, if successful, reduce the stress of employees. “Successful” is an important word in this context, for unsuccessful innovations, no matter their goal, are likely to increase stress. Obviously there will be successful innovations that make life better, and these will not be susceptible to the negative impacts we discuss below. We return to this in the discussion, but in the model and the analysis that follows, we focus only on innovations that increase stress. 2.2. Stress and productivity In a broad sense, stress can be understood as a measure of arousal; it is the result of a discrepancy between a desired and an actual situation, activating an actor to reduce this discrepancy (Selye, 1956 or Cox, 1978).1

1 In a more narrow sense, stress is related not to demands in general but to the inability to meet demands (e.g. Karasek, 1979); this is also referred to as ‘distress’. In this paper we adhere to the broader, stimulus-based definition of stress.

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The relationship between an agent’s performance in a given task and his or her level of “activation” or “arousal”, reflected in his level of stress, is generally thought to be curvilinear. The “Yerkes–Dodson law” (Yerkes and Dodson, 1908) argues that it is an inverted-U. Too little activation (low stress) implies lethargy and boredom, leading to low performance. Too much activation (high stress) implies an over-taxation of abilities, and again, under-performance. Intermediate levels of activation provide enough stimulus to alleviate boredom (and hopefully spark interest) without over-taxing the coping abilities of the actor. In line with its intuitive appeal, an inverted U shape relationship is generally confirmed in empirical research.2 The biological basis seems to be that up to a certain level stress hormones are effective in preparing body and mind for action (increased heart rate, more focused attention, etc.), but at high levels they have a negative effect on agents’ abilities to plan, remember, reason and regulate emotion (e.g. McEwen and Sapolsky, 1995; Sapolsky, 1996; Liston et al., 2006; Radley et al., 2006). 2.3. Stress and coping The physiological role of the stress response is to activate the agent to deploy his resources to deal with emerging demands (e.g. Selye, 1956; Karasek, 1979). Stress activates coping behaviour, which, when effective, reduces or eliminates the stressors. But at high levels of stress judgement deteriorates, behaviour increasingly gets misdirected, and this impedes an agent’s ability to produce effective coping (McEwen and Sapolsky; Sapolsky; Liston et al.; Radley et al.).3 2.4. Stress and social relationships: buffering effects Individuals involved in social relationships, with a spouse or colleague for example, have a more complex stress dynamic than those who are not. Being in a social relationship is generally beneficial, increasing well-being and reducing stress. Being in a relationship generally absorbs stress and has important buffering effects, which depend on the level of responsiveness of the individuals to each others’ needs. Social support from spouses, friends, colleagues and family can help to reduce stress and psychological strains (e.g. Glowinkowski and Cooper, 1985), and is one of the main mechanisms of “buffering” (e.g. Cohen and Wills, 1985; Lepore, 1992; Florian et al., 2002). 2.5. Stress and social relationships: stress spillovers Relationships can be beneficial in reducing stress, but they can also be a source of stress themselves. The stress level of one agent, through a variety of pathways, affects the stress level of the other. Because the internal stress system is non-specific, stress in one domain (work, for example) can “cross over” to another (home) (e.g. Hammer et al., 2005; Stephens et al., 1997). There is some compartmentalization between a person’s different roles, so a person can keep different parts of his or her life to some extent isolated, but this is never complete. Thus events in one domain will have an effect in another domain. (e.g. Bolger et al., 1989). But interestingly, stress spillovers from work to home are generally larger when work is high-skilled, when a job carries high responsibility and decision power, and when the worker is highly involved in his or her work (Ginn and Sandell, 1997; Glowinkowski and Cooper, 1985). This is the first step in the path linking innovation to stress in “unaffected” people. The second step is that stress also spills over between persons. This phenomenon is referred to as crossover and contagion. Crossover refers to the process that occurs when a psychological strain experienced by one person affects the level of strain of another person in the same social environment (Westman, 2001; Westman and Etzion, 1995). Contagion is the process by which one individual’s mood and/or perceptions seem to “spread” to those in close proximity. Hatfield et al. (1994) survey the large body of empirical evidence on crossover. 2.6. Responsiveness Both social support and stress spill-over are found to follow patterns of affiliation, and the strength of these processes depends on the strength of the affiliation (Gump and Kuliks, 1997). In essence, the extent to which relationships have either buffering or spillover effects on stress depends on the intensity of the relationship, but this intensity is itself affected by the stress levels of the actors involved. The quality and amount of social support a person is able to give is found to be negatively affected by the stress level of the giver (see Procidano and Smith, 1997, or Silverstein et al., 1996 for example). Repetti (1989) and Repetti and Wood (1997) for example, observe that an increase in a person’s stress level diminishes responsiveness to family

2

Though not universally e.g. Neiss (1988). In the absence of effective coping, the external stressors will not be reduced or eliminated, stress will increase and overall performance levels will drop, resulting in further stress. Additional stress will then lead to an increase in coping effort but a decrease of actual effective coping (e.g. Hobfoll, 1989). In order to keep our model tractable, we will use a linear coping function, and enfold this first derivative effect into the inverted-U relationship between stress and productivity. 3

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members. The introduction of stress into a relationship seems to reduce the ability of that relationship to provide buffering. At the same time, the extent to which spill-over between agents in a relationship takes place also depends on the intensity of the relation between the agents. Again this can be characterized as an issue of responsiveness. Several studies have found that cross-over and contagion effects are more likely to occur when people pay attention to, care for, identify with, or feel responsible for others (for example Benazon and Coyne, 2000). But the extent to which persons are responsive to each other is not constant; it can diminish as stress levels go up. Smith and Zautra (2001), for example, find that interpersonal responsiveness or sensitivity is found to increase reactivity to stressful events, especially those that are interpersonal in nature (see also Conger et al., 1999; Westman et al., 2004). Many empirical studies have shown that the higher a person’s stress level, the more maladaptive his relationship with his partner, the less other-focused and the more self-focused his behaviour will become. In other words, the responsiveness of persons to each other is a negative function of their stress levels (see for example Vinokur et al., 1996; Conger et al., 1999; Davila et al., 1997, 2003). To summarise this section briefly, we conclude the following: innovation intensifies the pressure of stress on the working population and affects productivity; individuals can reduce stress to some extent by activating coping mechanisms; individual occupational stress crosses to other domains and other individuals; social support helps to buffer individual stress; both responsiveness and efficiency of buffering depend on the strength of relationships; the quality of a social relationship is negatively affected by the level of stress in the relationship. In the next section we formalize these relationships. 3. Model In this section we examine the relationships between the innovation rate, the intensity of interpersonal relationships, and productivity growth. For that we set up and analyze a simple version of Aghion and Howitt’s (1992) growth model with fixed labour inputs. Here we bring together the key relationships discussed in the previous section. 3.1. Productivity and economic growth Consider a simple two-sector “quality ladder” model with an exogenously given rate of technical change in the R&D sector, and a fixed labour force distribution across the two sectors. In particular, assume that the number of workers in the manufacturing sector is NM , and in the R&D sector it is NRD . Without loss of generality in what follows we set NM = NRD = 1. Of concern, however, is not the number of workers, but rather efficiency units of labour, LM and LRD , defined by Li = l · Ni , where i ∈ {M, RD}, and l is the efficiency of labour, defined below. The manufacuring sector produces a final good with linear technology: Y = ALM ,

(1)

where LM is the labour input measured in efficiency units and A is the factor describing the state of technology, for example the stock of relevant technological knowledge. Technical change in the manufacturing sector is determined by the output of the R&D sector, , and the current stock of technological knowledge: dA = A. dt The R&D output, , is proportional to the labour input in the R&D sector, LRD , and some exogenously given innovation rate in the R&D sector, m  = mLRD . Assume now that the efficiency of labour is a function of stress levels. In accordance with the discussion in the previous section we assume an inverted-U relationship between labour productivity, l, and stress z (Yerkes–Dodson Law) l = l(z), where l(z) has an ‘optimum stress’ level zL∗ : l (zL∗ ) = 0, l (zL∗ ) < 0. Differentiating (1) and substituting, we see that the rate of economic growth, (m), is (m) ≡

1 dA 1 dl dz 1 dY = + . Y dt A dt l dz dt

In the long-run steady state of the system (defined by a constant stress level, dz/dt = 0) the second term related to the effect of stress on labour efficiency in the manufacturing sector is zero. Thus, since dA/dt = A, the steady-state rate of economic

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growth can be written as (m) = m · l(z(m)),

(2)

where labour efficiency, l, is determined by the level of stress in the R&D sector, z, which in turn is a function of the (exogeneously given) rate of technical change in the R&D sector. Eq. (2) is the same as equation G of Aghion and Howitt (1998, p. 60).4 Observe that growth rate (2) is determined by the Yerkes–Dodson law in the R&D sector, but not by the Yerkes–Dodson law in the final goods sector. In that sector, labour is subject to innovation and attendant stress, namely that arising as output from the R&D sector. Because in the long run the stress level in the final goods sector, caused by innovation there, adjusts (through the dynamics described below) and reaches a steady state level, it does not affect the productivity growth rate in the long run. It does, however affect the output level, as can be seen from (1). In order to understand the relationship between innovation in the R&D sector and economic growth, we first analyze the dynamics of innovation-induced stress. Since in the steady state the stress level of workers in the manufacturing sector does not affect long-term economic growth, in what follows we focus solely on the R&D sector. 3.2. Agents The population is composed of groups of individuals engaged in social relationships (e.g. organizations, families, neighbourhoods). Because our interest lies in the role that social relationships play in the dynamics of the stress, we classify the sources stress into those internal to the relationships among the members of the group, and those that are external. Therefore individual i belonging to a group of size n experiences stress both due to the (n − 1) social relationships he has with other members of the group, and due to factors external to those interactions. We assume a single source of external stress, innovation, which can affect different agents differently.5 Assume a positive, monotonic relationship between the rate of innovation, m, and individual stress levels from external sources: Si = Si (m) with Si (m) > 0. We describe the state of the individual i by his total stress level si . Individuals handle part of their stress via self-control, and we define c as the rate at which stress levels fall due to autarchic coping. But another way to reduce stress is to share it with the others. 3.3. Relationships The empirical studies cited above suggest that the stress spill-overs and stress buffering are functions of the quality of relationships between individuals. At the same time the strength of a relationship depends on the level of the stress within the relationship. For simplicity and tractability, let us assume that all relationships between the members of a group are of the same strength n and depend only on the average stress (z = (1/n) i=1 si ).6 Formally, we introduce the parameter a ∈ [0, 1] describing the strength of the social bonds between the members of the group: a = 1 corresponds to a normal functioning relationship, while a = 0 corresponds to the situation in which all social relationships have completely broken down. As additional stress in a relationship is damaging to the relationship, we assume that a (z) < 0, with a(z) twice continuously differentiable. When there is no stress, relationship strength is at its maximum (a(0) = 1); as the stress level increases, relationship strength decreases and the relationship eventually ceases (lim z→∞ a(z) = 0). Sharing stress has two effects. First, social support from the other members of the group reduces individual stress. Define b(a) as the buffering function, the rate at which individual stress is reduced by sharing stress with the others. And assume that b (a) > 0 (b(a) is twice continuously differentiable on [0,1]). We also assume that the existence of social relationships is a prerequisite for buffering to be activated (b(0) = 0), and the rate of buffering is finite b(z) < ∞ for all z. Second, due to the spillover effect discussed in the previous section, sharing stress with one’s partners causes them stress and again the intensity of this effect depends on the strength of social relationships in a natural way. Defining p(a) as the spillover function we assume that 0 < p(a) < ∞ , ∀ a ∈ [0, 1]; p(a) is a twice continuously differentiable increasing function of the relationship strength with p (a) > 0; and the absence of a relationship implies no spillovers (p(0) = 0).

4

Also compare (2) with Eq. (3.11) of Aghion and Howitt (1992). Anticipating the analysis of Section 4.2 we here treat innovation as the sole source off external stress. This is obviously generalizable: any source of stress external to the relationship (financial, personal, general job stress, employment uncertainty . . .) will drive the same dynamics of stress within the relationship. 6 More generally one could allow the relationships between  different pairs of individuals vary in strength and introduce an “adjacency matrix” to describe 5

the structure of interactions within the group: A = ai,j , where ai,j characterizes the strength of the relationship between agents i and j.

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3.4. Dynamics of stress We formalize the dynamics with the following system of differential equations.

⎧ ds1 ⎪ ⎪ ⎪ dt = −c · s1 − b(a) · s1 + p(a) · s2 + · · · + p(a) · sn + S1 , ⎪ ⎪ ⎪ ⎪ ⎨ ds2 = −c · s − b(a) · s + p(a) · s + · · · + p(a) · s + S , n 2 2 1 2 dt ⎪ ⎪··· ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dsn = −c · s − b(a) · s + p(a) · s + · · · + p(a) · s + S , n n n 1 n−1

(3)

dt

where the strength of social relationships within the group, a, is a function of the average stress in the group: a = a(z). According to (3) individual i can reduce his stress, si , through the mechanism of self-control (at the rate c) and by sharing stress with his partners (at the rate b(a)). At the same time i’s stress level increases due to spillovers from his partners: p(a)sj . Summing Eqs. of (3) and dividing by n we can write the dynamics for the average stress, z, as dz = −cz − b(a)z + (n − 1)p(a)z + S0 , dt

(4)



where a = a(z), and S0 ≡ (1/n) Si is average external stress.7 Average external stress is a function of innovation S0 = S0 (m). In what follows, we leave the level of individuals, and treat the group as a whole as the unit of analysis. 4. Analysis 4.1. Dynamics of stress To clarify the roles of different forces in the model we begin by discussing individually the roles of coping, buffering, and responsiveness in the dynamics defined by Eq. (4). 4.1.1. Coping Suppose there is no relationship (a = 0), so that the second and the third terms of (4) are equal to 0. Then Eq. (4) is simply dz = −c · z + S0 . dt

(5)

The phase diagram corresponding to (5) is a straight line with vertical intercept at S0 and negative slope −c. The only steady state of (5) is at zc = S0 /c. The steady state is stable, since at this level of stress dz/dt changes its sign from positive for z < zc to negative. It implies that as long as the long-run values of the parameters (external stress, S0 , and self-control, c) stay the same, the system always returns to the long-term level of stress, zc , regardless of the magnitude of any disturbance that takes the system away from its steady state. Better coping abilities correspond to a steeper slope and reduce the steady state stress level, zc , while an increase in the inflow of the external stress (related to innovations) shifts the line upwards and increases zc . Furthermore, (5) implies that an increase of 1 percent in external stress leads to a 1 percent increase in the steady state stress level, zc (similarly for coping). 4.1.2. Stress spillovers Consider the system (4) without stress absorption mechanisms, that is, with neither buffering nor coping (b(a) = 0, c = 0). Then the system’s behaviour is determined by the dynamics of stress spillovers alone: dz = (n − 1)p(a(z)) · z + S0 . dt

(6)

Stress spillovers are positive for any non-zero z: (n − 1)p(a(z))z > 0, and become zero only at z = 0 (no stress spills over when no one is stressed). The inflow of external stress is always non-negative (S0 ≥ 0). Hence the right hand side of (6) becomes zero only when S0 = 0 and only at z = 0; otherwise it is always positive. Thus the system has a unique (unstable) steady state with zero stress level only when there is no inflow of external stress. Any (even temporary) disturbance would lead to accumulation and magnification of stress due to interpersonal spillovers of stress within the group.

7 System (3) has non-zero determinant when c + b(a)z = / (n − 1)p(a). Thus when Si = S0 > 0 for all i, all agents experience the same level of stress in the steady state (the system has a unique symmetric steady state for a given z).

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4.1.3. Buffering Consider Eq. (4) with neither coping (c = 0) nor stress spillovers (p(a) = 0): dz = −b(a(z)) · z + S0 . dt

(7)

The dynamics of the system are determined by the shape of the function b(a(z)). Notice that b(a(0)) = b(1) and therefore b(a(z)) · z = 0 at z = 0. Let us assume that lim b(a(z)) · z = 0,

(8)

z→∞

that is, for high levels of stress the relationship weakens to such a degree that buffering goes to zero. Then we can state the following: ¯ the dynamic system defined by (7) (which includes only buffering) Proposition 1. There exists an S¯ < ∞ such that for S0 ∈ (0, S) has at least two steady states. Proof. The function b(a(z)) is a composition of twice continuously differentiable functions a(z) and b(a), and thus is itself twice continuously differentiable. So is b(a(z))z. Continuity of b(a(z))z together with condition (8) ensures that b(a(z))z is ¯ at some finite z∗ (if there is more than one point bounded from above and it reaches its maximum value, which we denote S, ¯ let z∗ be any of them). at which b(a(z))z = S, ¯ The zeroes of (z) determine the steady Define the function (z) as the right hand side of the Eq. (7) and let S0 ∈ (0, S).  ¯ there is at least one value = 0, and b(a(z))z  ∗ = S, states of (4). Taking into account that b(a(z))z is continuous, b(a(z))z  z=0 z=z z1 ∈ (0, z∗ ) such that b(a(z1 ))z1 = S0 and therefore (z1 ) = 0. Condition (8) implies that for any Sm ∈ (0, S0 ), there is a value zm (Sm ) such that b(a(z))z ≤ Sm for any z ≥ zm . Since b(a(z))z is continuous, b(a(z ∗ ))z ∗ = S¯ > S0 and b(a(zm ))zm < S0 there is at least one value z2 ∈ (z∗ , zm ) such that b(a(z2 ))z2 = S0 and therefore (z2 ) = 0 (in case of many values like that let z2 be the largest of them). Thus Eq. (7) has at least two steady states z1 and z2 .  Also notice that the steady state with the highest level of stress, z2 , is unstable because (z) > 0 for z > z2 . Any positive disturbance would set the system on the path of increasing stress and decreasing relationship quality. Beyond a certain threshold value of stress, a social relationship can no longer function as a stress-absorbing mechanism. 4.1.4. General case Now let there be coping, buffering and stress spillovers. The dynamics of the system defined by Eq. (4) includes these three factors, and gives them the characteristics discussed in Section 2 above. Their relative strengths determine which steady state is realized. There is a number of the possible combinations of steady states, and in this paper we limit our analysis to one case with interesting dynamics that allows an intuitive interpretation. For that collect all of the social effects of stress (that is buffering and spillover) into a single function: R(z) = b(a(z)) − (n − 1)p(a(z)). First, assume that agents are in general better off having social relationships rather than dealing with stress on their own (the positive effect of buffering exceeds the negative effect of stress spillovers)8 : R(z) > 0 for z > 0,

(9)

and assume that the effects of relationships weaken as stress in the relationship grows (because the strength of the relationship decreases): R (z) < 0 for z > 0.

(10)

Second, assume that the effects of having social relationships disappear as z tends to infinity (because a social relationship loses its strength as the stress level in the relationship increases): lim R(z)z = 0.

(11)

z→∞

Third, assume that social effects become less responsive as stress levels increase. Specifically, defining R as the elasticity of R(z) with respect to stress, we can write dR (z) ≡ dz

R (z)  z

R(z)

< 0.

(12)

8 One can easily imagine that not all relationships are stress-reducing on net. We assume that stress is sufficiently important in the agents’ utility functions that rational agents will exit any relationship that is, on net, stress inducing. This justifies the assumption that R(z) > 0.

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0.5 S0 − (c+b−r)z S0 − cz

0.4

zero line 0.3

dz/dt

0.2 0.1 0

z2

z1

z3

−0.1 −0.2

0

5

10 15 Average stress, z

20

25

Fig. 1. Phase diagrams for Eq. (4). An example of a phase diagram for the system dynamics (using a = e−z , p(a) = a, and b(a) =

√

a; c = 0.03, S0 = 0.5).

In addition to (12), assume the same for the elasticity of the marginal effect of the social relationship R (z): dR (z) ≡ dz



z

R(z)



R(z)

< 0.

(13)

Now we can state Proposition 2. Under conditions (9 – 13), there is a critical coping value, c¯ , such that for any c ∈ (0, c¯ ) there are S1 and S2 such that depending on the external stress S0 , the system (4) follows one of the three qualitatively different dynamics: (1) S0 ∈ [0, S1 ). There is single stable steady state z1 . (2) S0 ∈ (S2 , ∞ ). There is single stable steady state z3 . (3) S0 ∈ (S1 , S2 ). There are three steady states z1 < z2 < z3 . States z1 and z3 are stable, z2 is unstable. Proof.

See Appendix A.



The proposition is worth a few words of explanation. The system’s behaviour is largely affected by the relative strength of coping versus buffering and spillovers. In groups of individuals with superior coping abilities (c 1) the former dominates the latter and consequently the dynamics is qualitatively similar to the case of no relationship (5) analyzed at the beginning of this section. However in the more realistic case of small and intermediate values of c (c < c¯ ) the system’s dynamics combines features of both (7) and (5).9 A typical phase diagram corresponding to such a case is shown in Fig. 1, using the √ following functional forms: a = e−z , p(a) = a, and b(a) = a. The solid curve in Fig. 1 corresponds to the reaction function dz (z), and the dashed line represents the reaction function dt without the relationship (5) for the same values of c and S0 . For small and intermediate values of z a social relationship works as a stress absorber: the solid line is far below dashed line. However as stress increases, the relationship starts to fade away and can no longer absorb stress: the reaction function approaches the no-relationship case. According to the proposition the system may have multiple steady states. Fig. 1 presents an example of such a system with three steady states: z3 is similar to the case of no relationships (5) while z1 and z2 arise due to the buffering function of social relationships (7). 4.2. Innovation and stress The relationship between innovation and stress follows directly from Proposition 2. The intensity of innovation (other things equal) determines the value of external stress S0 . Without loss of generality we can assume that the stress level is equal to the rate of innovation: S0 = m. Analysis is simpler if in the dynamics (Eq. 4) we isolate exogenous from endogenous effects on stress. To capture endogenous effects define a function f(z): f (z) = −(R(z) + c)z.

9

Under the assumptions of the proposition spillovers do not add any steady states.

(14)

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S0=0.3

0.6

S0=0.5 S0=0.7 zero line

0.4 Low− and high− stress steady states coexisting

dz/dt

0.2

0

−0.2 High−stress (the only) steady state −0.4 Low−stress (the only) steady state 0

5

10

15

20

25

Stress, z Fig. 2. Phase diagrams for Eq. (4). Effect of external stress.

This function has two local extremes at z1∗ and z3∗ : z1∗ < z3∗ , defined by: f  (z1∗ ) = f  (z3∗ ) = 0,

(15)

where z1∗ is a local maximum, and z3∗ is a local minimum of f(z) (see the proof of Proposition 2). Depending on the external stress the system may have three qualitatively different dynamics: for a low innovation rate m: m < S1 ≡ f (z3∗ ) there is a unique low-stress steady state (stable); when the innovation rate is high: m > S2 ≡ f (z1∗ ) in the long-run the system always converges to high-stress steady state (stable); under intermediate rate of innovation S1 ≤ m ≤ S2 low- and high-stress equilibria co-exist. The three dynamics corresponding to different ranges of the innovation rate, or external stress, are shown in Fig. 2. • Low innovation intensity (dashed line). There is a unique steady state with relatively low stress (as compared to the situation of no relationship): the social relationship is effectively dissipating stress. • High innovation intensity (solid grey). S0 is large; there is single high-stress equilibrium. The social relationship is counterproductive and has become a stressor in itself. The relationship therefore will fade out (high stress, z, means low responsiveness, a). In the case of a family relationship, we may think of divorce, since the partners could not cope with the ever growing stress; in case of work relationship, colleagues avoid each other, or do not talk to each other. Each of the partners has to rely on individual self-control mechanisms. This case is similar to that of no relationship. • Intermediate innovation intensity (solid black). Two equilibria co-exist (the system has two stable equilibria and one unstable equilibrium). Depending on the initial conditions, we have either a low-stress equilibrium where the social relationship is absorbing the stress (as in the low innovation intensity case), or a high stress equilibrium with (almost) no social relationships (only self-control, similar to high innovation intensity). The last situation is particularly interesting. In this case the system has memory. Suppose that the system is in a low-stress equilibrium with social relationships buffering stress. A temporary shock to S0 (a major disruptive innovation episode, for example) will shift the dz/dt curve up, and if the magnitude of the shock is sufficiently high, it can make the system ‘jump over’ the unstable steady state to the high stress equilibrium. Even if the external stress returns to the initial value of S0 , the system will continue to reside in the high-stress equilibrium. Thus, in this case relatively small changes in external stress S0 may lead to drastic qualitative changes in the behaviour of the system, and changes that are difficult to reverse.10 The effects of external stress caused by innovation on the social relationships in the group and on the average level of stress are summarized in Fig. 3. The left panel shows the relationship between external stress and the strength of social relationships. When there is no inflow of stress, there is no stress in the system. At this point the group is tightly bound with strong social ties (a = 1). As external stress grows, the quality of social relationships deteriorates, but social support from one’s partners continues to help reduce the stress. Small temporary shocks may drive average stress in the group away from the steady state corresponding to the given value of external stress, however with time the system always returns to the corresponding steady state that is located on the dashed line of Fig. 3a.

10

In terms of dynamic systems theory our system has a “cusp” catastrophe (Saunders, 1980).

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a

b

z1 (a ~ 1)

0.9

z2 (unstable)

0.8

z3 (a ~ 0)

30 z (a ~ 1) 1 z2 (unstable) z (a ~ 0)

25

0.7

3

20

Stress, z

Relationship strength, a

1

0.6 0.5

15

0.4 10

z*3

0.3 0.2

5 z* 1

S1

S2

0.1 0 0

S1 0.2

S2 0.4

0.6

External stress, S0

0.8

1

0 0

0.2

0.4

0.6

0.8

1

External stress, S0

Fig. 3. Left: Strength of social relationships as a function of external stress. Right: Average stress level as a function of external stress.

The first drastic change happens if the intensity of external stress reaches S1 . At this point another stable steady state, corresponding to z3 in Fig. 1, appears. This steady state is characterized by extremely weak social relationships (a ≈ 0)—social relationships essentially vanish, at least as sources and sinks of stress. The long run behaviour of the system undergoes one more qualitative change when the intensity of external stress approaches S2 . At this point the steady state with social relationships disappears and only the steady state with no social relationships continues to exist, that is to say, the inflow of external stress with an intensity exceeding S2 destroys the relationships between the members of the group. The corresponding evolution of the long-run level of average stress is shown in Fig. 3b.11 Between S1 and S2 the two steady states co-exist. In which of the steady states the system resides is determined by the history of the system. The relationship between innovation and stress levels also shows hysteresis. If the innovation rate rises above S2 , the system is on the upper branch of the stress curve. Reducing the innovation rate from that level does not immediately jump the system to the lower branch. Instead, it will rest on the upper branch until the innovation rate is reduced below S1 , at which point it will jump down again to the lower branch. The explanation for the hysteresis lies in the fact that above a certain level of external stress, relationships themselves start to become stressors and will eventually break down. If this happens, even if the level of external stress is brought back to intermediate levels again, the buffering effects of relationships are no longer present. The cycle is vicious. The absence of the buffering effect of relationships makes it more difficult to restore relationships (increase responsiveness), and this maintains the stress that is “internal to the relationship” at a high level. Thus only for much lower levels of external stress will responsiveness recover, and return relationships to their stress-reducing role.

4.3. Innovation, stress, and economic growth Economic growth is defined above in Eq. (2) as (m) = m · l(z(m)), where m is the (exogenous) innovation rate in the R&D sector, and z(m) is the associated stress level. Innovation has two effects on economic performance. First, there is a positive direct effect derived from the fact that innovation brings new, and more efficient methods of production, thereby increasing labour productivity. But there is also an indirect effect related to the fact that innovation increases the flow of stress, as discussed in Section 2.12 According to the Yerkes–Dodson law this will change labour productivity, and the direct and indirect effects may act either in the same or in opposing directions. We show here the possibility that the negative indirect effect can overwhelm the positive direct effect, depending on the initial level of stress. The relationship between innovation and long-term average stress level can be derived from (4), (14) and the Yerkes–Dodson law. Assuming that in the long-run the system resides in a stable steady state with an unchanging stress level (dz/dt = 0), f (z) = m for z : f  (z) > 0,

11

The plot is provided by the equation: S0 = f(z) (Eq. 14). We should recall here that not all innovations are stress-inducing. Indeed there are innovations, often of the re-organization type, that are specifically designed to reduce stress. The analysis applies directly to innovations that are stress-inducing. 12

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where the condition on f (z) ensures stability. Since for m ∈ (S1 , S2 ) two stable steady states coexist, and we need two functions Z1 (m) and Z3 (m) for low- and high-stress equilibria respectively to invert Eq. (14): Z1 (m) = f −1 (m)

for

z ≤ z1∗ ,

Z3 (m) = f −1 (m)

for

z ≥ z3∗ ,

(16)

where z1∗ and z3∗ are defined in (15). Inserting (16) into (2) we obtain the relationship between the rate of innovation and economic growth at the two possible steady state levels of stress: 1 (m) ≡

1 dy1 = m · l(Z1 (m)), y1 dt

3 (m) ≡

1 dy3 = m · l(Z3 (m)), y3 dt

(17)

where dy1 /dt is economic growth in a low-stress society bound by social ties, while dy3 /dt relates to a population of stressed individuals ‘bowling alone’. It seems reasonable to assume that the high-stress, no-relationship equilibrium of our model is located above the ‘optimum stress’ of the Yerkes–Dodson law (zL∗ ), on the descending part of the inverted U. In other words we assume that for an individual who is stressed to such a degree that (s)he is unable to support a long-term social relationship, a further increase in stress will decrease his(her) productivity. Similarly we assume that the low-stress equilibrium is at or below the ‘optimum stress’.13 Furthermore, we require that in the high-stress equilibrium productivity, l(z) is more elastic than f(z), i.e, −l > f . Under these conditions we can state Proposition 3. The effect of innovation on economic growth defined by (17) depends on whether the system resides in the highor low-stress steady state: ∂1 > 0, ∂m Proof.

or

∂3 < 0. ∂m

Differentiating (17) with respect to m gives us ∂1,3  = l(Z1,3 ) + m · l (Z1,3 )Z1,3 (m). ∂m

By assumption for all z < z1∗ : l (z) > 0. Therefore ∂1 = l(Z1 ) + m · l (Z1 )Z1 (m) > 0. ∂m Since for all z > z2∗ : l (z) < 0 and −l > f ∂3 = l(Z3 ) + m · l (Z3 )Z3 (m) = l · ∂m

 1+

l f

< 0.

 Fig. 4 illustrates the relationship between innovation and economic growth when labour productivity in the high-stress equilibrium is lower than productivity in the low-stress steady state. For low innovation rates, (small m), and accompanying low levels of external stress (S0 ), and thus low steady state stress levels, economic growth responds rapidly and positively to an increase in the innovation rate. By contrast, at high innovation rates, high external stress and high steady state stress levels, an increase in the innovation rate has a negative effect on the growth rate. Here the secondary effect of innovation, related to its role as a stressor on workers in the R&D sector, overwhelms the positive effect and productivity falls in response to increases in innovation. The co-existence of equilibria in the middle range of the innovation rate makes an economy unstable with respect to small fluctuations in the flow of external stress—small shocks may flip the system out of the low(m). We should emphasize that these small shocks may originate stress equilibrium and set it on the declining branch of dy dt in sources completely unrelated to innovation. To return the system to the low-stress equilibrium we have to decrease the innovation rate much lower (so that only the low-stress equilibrium exists). In other words, not only do we find multiple steady states, we also find that the hysteresis present in the stress relation carries over to the relationship between innovation and productivity. For an intermediate range of innovation rates there is a double value function. Jumping from one branch to the other, as the innovation rate passes a threshold has the character of a phase-change. If after crossing this threshold from below, say, two things change. The hysteresis implies that if innovation is reduced again, z will remain on the high value segment of the stress curve. If innovation rates become too high, thereby

13

Formally z1∗ < zL∗ < z3∗ , where z1∗ and z3∗ are the maximum stress in low-stress state and minimum stress in high-stress state respectively (see Fig. 3b).

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4 3.5

Growth rate

3 2.5 2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

Innovation rate, m Fig. 4. Economic growth as a function of the innovation rate in R&D sector (a = e−z , p(a) = a, b(a) =

√

a, and l(z) = 4z · e−z/4 ; c = 0.03, S0 = 0.5).

creating high rates of stress flow, a reduction to the (previously maintained) intermediate level of innovation and hence external stress, we will nevertheless retain a much higher stress than before. In order to lower stress levels, we will have to lower the innovation rate considerably below the original rate which was initially associated with acceptable stress levels. The second effect of this phase change is that the relationship between innovation and growth changes sign, due to the Yerkes–Dodson law. We move from a situation in which increasing innovation increases growth, to one in which the reverse is true. Increasing the innovation rate thus can induce a decline in productivity, including productivity of those producing innovations. And this, of course, passes through to economic growth. The policy-maker faces a double bind. The obvious policy to increase growth, namely increasing the rate of innovation, is more difficult to obtain, since those producing innovations have lower productivity. And if, nonetheless (perhaps due to a re-allocation of resources into innovation production) the intermediate goal, increasing innovation, is successful, the ultimate goal, increasing growth, will not be. In fact, from this position to increase growth, a policy-maker should decrease the innovation rate in order to reduce stress levels and increase productivity. And in doing so he receives a double effect. As with the stress relationship, for intermediate values of the innovation rate, whether or not increasing it will increase growth depends on whether we are on the upper or lower branch, which depends again on history. Ironically, this implies that it is economically costly to have innovation rates that are extremely high, even if only temporarily. The price is that due to the breakdown of relationships people are less able to reduce their stress levels and therefore are less able to deal with higher intermediate rates of innovation than they were before. 4.4. Parameter effects In this section we are interested in the consequences of of two parameters, namely coping, and buffering. This will give some perspective on possible interventions and their effects. 4.4.1. Coping skills Coping refers to an individual’s autonomous ability to reduce stress, and is captured with the parameter c.14 In the absence of a relationship, if c is increased, an individual reduces his stress level at a higher rate, and long-term levels of stress go down. Given a fixed stream of new innovations, he will arrive faster at a steady state stress level, and as we observed in Section 4.1, that steady state stress level is negatively related to c. We see a similar effect when we look at people engaged in social relationships, however in this case a change in coping abilities may have much more dramatic effect. An example is shown in Fig. 5 wherein a small increase in c has two effects15 : it reduces the value of stress in the high level equilibrium, and it creates the low level equilibrium. The latter occurs because coping can bring stress levels down to the point where social relationships may effectively buffer external stress. Thus increasing coping skills results in a lower steady state of stress, especially in the presence of a relationship. Conversely

14 Outside the model, and outside this discussion, is the fact that accumulation of stress over time may negatively affect one’s self-control reducing abilities to cope with stress. In principle this could be included in the model, but would only sharpen the results. For an examination of how individuals learn better coping skills see for example Greve and Strobl (2004). √ 15 The figure is drawn using: a = e−z , p(a) = a, b(a) = a, and l(z) = 4z · e−z/4 ; c = 0.03, S0 = 0.5.

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c=0.026 c=0.029

0.4 0.3

dz/dt

0.2 0.1 z (c=0.026) c

0 z1(c=0.029)

zc(c=0.029)

−0.1 −0.2

0

5

10

15

20

25

Stress, z Fig. 5. Phase diagram for two values of coping: c = 0.026, black curve; c = 0.029, grey curve.

a deterioration in self-control may have additional adverse impact in the presence of social relationships if it leads to a breakdown of the relationships, as beneficial effects of buffering go away together with the social relationships. 4.4.2. Buffering The quality of a relationship enters the model through buffering: b(a). Since b(a) is a product of a compound variable including social skills and reflection, then increasing social skills and reflection will shift b(a) upward, increasing the potential beneficial effects of having a relationship. Holding spillover levels (p(a)) constant, the net result of being in a relationship will be more positive and will remain positive longer, at higher levels of stress, than before. This implies that both the low and the high steady state of stress level z are located at lower levels of stress. Quantitatively, shifting b(a) upwards by a constant, b0 , has exactly the same effect as increasing c by b0 , therefore the effect of change in the quality of the relationship is equivalent to the effect of the corresponding change in coping abilities analyzed above (Fig. 5). 5. Discussion and conclusions In this paper we have presented a simple model of induced stress. Innovation is, rather schematically, taken as a source of stress, because it leads to change and uncertainty. While an agent is exposed to stress resulting from the demands of his environment (such as innovation), the stress level he ultimately experiences is mediated by aspects of a relationship between himself and another agent. Associating stress with the Yerkes–Dodson inverted-U permits us to take this micro-model of stress contagion as a basis for understanding how innovation and growth interact with mental capital. The model implies that it is possible to get “too much of a good thing.” High levels of innovation, if not accompanied by increases in agents’ abilities to cope with their own stress, or buffer each others’ stress, may actually be counter-productive in terms of significantly reducing the net productivity growth resulting from innovation. In addition, at high levels of stress, the relationship between stress and productivity may exhibit hysteresis. If rates of change become so high that an individual, organization or society is tipped into the high-stress equilibrium, a simple re-establishment of the previous, lower rate will not necessarily re-establish the lower stress levels. Not only will it take time, but also a certain degree of ‘over-correction’ in order to ‘tip back’, which can be costly. There is now significant neurological evidence of hysteretic effects of stress at the level of the individual. Prolonged exposure to stress negatively affects the plasticity and reactivity of the brain (Sapolsky, 1996) and is related to neurological atrophy (e.g. Zaidel et al., 1996) and functional decline (McEwen and Seeman, 1999; Karlamangla et al., 2002). These alterations of the brain at the biological level are still present long after removing the stress, so recovery of functionalities can involve more than a simple return to lower environmental stress levels. Moreover, stress is found to cause cumulative damage to the part of the brain involved in managing stress, thus the capacity to recover from stress is itself diminished by stress (Sapolsky, 2004). The neuroendocrinological system of stress has a feed-forward cascading dynamics (Sapolsky et al., 1986), resulting in asymmetric effects of increasing and diminishing stress. For example Chemtob et al. (1988) find that the ‘activation threshold’ is at a higher level of stress than the ‘deactivation threshold’ which implies the possible existence of hysteresis within the individual. We also observe positive feedack between stress and the deterioration of stress-management systems (e.g. Chemtob et al., 1988), so at the individual level, the capacity to recover from stress is itself affected by stress. This is seen in the growing body of social–psychological research supporting a resource-based view of coping capability (Hobfoll, 1989), which

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seems to indicate an asymmetry of resource loss and resource gain. For example, Jansen et al. (2003) found that stress and especially the related prolonged fatigue lead to both elevated need for recovery and reduced opportunity for and effectiveness of recovery.16 Demerouti et al. (2004) provided evidence for “loss spiral” of work stress and performance; work pressure caused loss of personal coping resources, which evoked work family interference and then feelings of exhaustion; these feelings of chronic fatigue then resulted in more work pressure, starting the cycle over again. In line with this, the research on ‘recovery of work’ finds that standard recovery techniques such as taking a vacation are less effective at higher workrelated stress levels; high initial stress levels diminish the recovery effect of vacation, they reduce the ‘efficiency’ of vacation in terms of producing recovery (Fritz and Sonnentag, 2006). This implies that stress reduction gets more difficult at high stress levels, when it is needed most, and it implies hysteresis due to the reduction of stress recovery potential due to stress. While spillover and crossover effects have been well-documented (see Section 2), to our knowledge there has been no empirical study of hysteresis effects operating through social contacts, and still less on productivity effects. However, our findings resonate with a number of other findings, most importantly in the area of social capital, organizational change, and skills; they are in line with findings in these literatures, and in fact might offer additional insight on and explanation for these findings. Our results are in line with the findings on the (positive) effect of social capital on productivity, careers, competitiveness, innovation, and economic growth. Social capital refers to the presence of social relations and resulting phenomena such as trust (Durlauf, 2002), and research suggests that social capital tends to have a positive effect on economic outcomes (e.g. Knack and Keefer, 1997; Florin et al., 2003; Seibert et al., 2001). Currently, the main explanations given for this effect are that social capital reduces transaction costs (e.g. Knack and Keefer, 1997), leads to better access to information, resources and knowledge (e.g. Levin and Cross, 2004), and leads to increased motivation and organizational citizenship behaviour (e.g. Bolino et al., 2002). The effect of social capital seems to a considerable extent to be indirect, often via its effect on innovation (e.g. Akc¸omak and ter Weel, 2009; Zheng, 2008), and seems to be especially important in firms with high rates of innovation (e.g. Smith et al., 2005). In a recent review on the empirical findings on the relation between social capital and innovation, the social capital elements trust and relationship strength seemed to have the strongest positive relation with innovation (Zheng, 2008). While our results are in line with this other research, our model suggests an additional explanation for the positive effect of social capital, namely that social relationships reduce stress, thus reducing the negative productivity effects (due to stress) accompanying productivity-enhancing factors such as innovation. In the case of innovation, the result is a higher net effect of innovation in terms of productivity, and a higher level of innovation that can be tolerated and maintained by individuals and organizations.17 The notion of social capital as a stress-buffering resource is in line with empirical findings in the organization psychological literature; for example, Halbesleben and Bowler (2007) find that employees with emotional exhaustion due to high stress levels tend to shift their resources from actual work towards investing in social support aimed specifically at stress buffering. This implies that the relationship between social capital, innovation and productivity may have an important pathway via its effect of stress reduction. It also implies that an increased need for stress-buffering shifts resources (time, effort) away from directly dealing with stressors (such as tasks or work pressure) towards maintaining and increasing stress-buffering factors such as social relations, reducing direct productivity. The effect of social capital implied in our paper is different from the ruling interpretations in economic research at the moment, and might offer new venues for research on the economically relevant effects of social capital. Our results are also in line with the findings on the effects of investments in organizational structures and human resource practices. These seem necessary to realize the full potential of information technology (IT) investments, the main rationale being that they enable better use of the new ways of working which a new technology offers (e.g. Black and Lynch, 2001). Our model suggests an additional reason for the effect of such investments. Research suggests that in particular the introduction of flatter hierarchies, team work, and decentralization of decision-power and problem-solving to the levels of teams and individuals, substantially increase the return to IT (e.g. Brynjolfsson and Hitt, 2000; Lam, 2004). These types of changes increase the control of the individual employee over his work and thereby over his own stress level, which could provide an additional reason for the positive productivity effects of such organizational changes, and of the complementarity of these changes with innovation. In line with this, it seems that this effect is especially high in firms characterized by high rates of change, more complex processes, more innovation, or a more competitive environment (Shaw, 2004), that is, firms in which coping with stress might become a critical variable.

16 As stress levels increase, the personal resource management problem more difficult. Managing those resources itself consumes more resources, (see also Hockey, 1997). Thus stress shifts coping resources away from direct coping with stressors towards the regulation of diminishing coping resources, accelerating the process of loss of resources. 17 Some qualification is needed here. In the existing social capital literature, social capital is often measured as the strength of ties between employees within and between firms, or between employees and customers; in our model we focus on not only the strength but also the quality of dyadic relationships such as between spouses and close co-workers. Nevertheless, it seems fair to assume that less-close relationships also have stress-buffering capacity (thus the social capital research measuring these relations is still relevant), and to assume that there is a correlation between the quality of relationships: persons with better relationships in one domain also having better relationships in other domains, thus enabling one to use relationship quality of less close relationships as a signal about the quality of closes relationships (although there might be some degree of ‘trade-off’ as the weak–ties–strong–ties literature suggests, e.g. Hansen et al., 2001).

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Obviously, organizational change is also a change and thus a potential source of stress. One implication here is that both the size and the timing of investments in change may be very important. A combination of HRM changes and technological changes might have the best net-effect, as long as the total level of change and thus of stress is balanced out by the increased stress-reducing capabilities of employees. From the perspective of our model, it is important to carefully ‘dose’ innovation, to make sure that the (initial) stress-generating effect of innovation does not tip an individual or an organization over the threshold of its stress-coping abilities, thereby reducing the productivity enhancing effects of innovation, and disabling its potential stress-reducing effect. Indeed, a further implication is that the sequence of changes might be very important. A stress-increasing technical innovation, followed by a stress-reducing organizational change, may suffer from hysteresis effects which would not be present were the order revered. In line with this, there is evidence suggesting that a large part of all firm re-organizations fail to meet their objectives (some authors simply say “fail”), even when they are aimed at improving the organization rather than merely cost-cutting (e.g. CIPD, 2004; Van Witteloostuijn, 1999). This is odd, since major organizations can expect to undergo major restructuring on average every three years (CIPD, 2004). Employee-related issues emerge as the most problematical aspect of re-organisations, with staff complaints and turnover particularly likely to be increased. It seems likely that many re-structurings are responses to threats that the organization feels, thus signaling high stress situations. If the labour force of a firm is sitting in the high-stress equilibrium when the re-structuring takes place, even if the re-structuring is well-designed, and should reduce environmental stress levels, hysteresis effects may imply that these do not have the expected effects on productivity. This might help explain why successful re-structuring is so difficult to produce. Ultimately, what matters is not whether an innovation in itself is stress-inducing or stress-reducing, but rather the conditions under which it is so. From the persepctive of our model, a diminished effect or time lag of productivity effects might also be related to the limited ability of actors to cope with change, and the possibility of stress affecting the ability to cope with stress. This has the more the character of a resource constraint (Hobfoll et al., 1989) rather than a skill-constraint, and is in line with growing research findings on ‘loss spirals’ such as in Demerouti et al. (2004). In each of these examples, one can see a role for employee stress mediating economic outcomes. If this is the case, then understanding the dynamics of stress and how it interacts with productivity may be important in understanding the phenomena. The point of our model has not been to argue that including stress and stress-reducing capabilities shows that high rates of innovation are bad, but rather that including them increases the understanding of the complexity of the interaction between innovation, agents, stress, productivity growth, and ultimately welfare. Simon’s (1971) remark that a wealth of information implies a poverty of attention may apply here, and resonates with the demand and control model of stress. The effect of innovation on productivity is not simple or equivocal, but crucially depends on the coping capabilities of employees and on which type of stress-equilibrium employees and organizations as a whole are residing. A final implication of this is that the common practice of developing ever more policy aimed and increasing innovation in order to increase productivity and economic growth may be self-defeating, and worse, may create a hysteretic effect that forces an extended period of slow innovation and slow growth in order for growth to recover. Appendix A. Proof of Proposition 2. The dynamics of the system is greatly affected by properties of R(z). Therefore before we start with the proof of the proposition let us examine some properties of this function. First, notice that R(z) and its derivative are continuous functions of z. Indeed, function R(z) defined as R(z) = b(a(z)) − (n − 1)p(a(z)), where b(a), p(a), and a(z) are twice continuously differentiable, is twice continuously differentiable function itself. Furthermore, conditions (9) and (10) imply that R(z) is strictly positive monotonously decreasing function of z. Second, the elasticities of R(z) and R (z) are continuous monotonous functions of z. The elasticity of R(z) defined as

R (z) =

zR (z) R(z)

is a ratio of continuous functions with the denominator different from zero (strictly positive), and hence is a continuous function of z. By assumption (12) R (z) is monotonously decreasing. Similarly R (z) as a ratio of continuous functions with nonzero denominator (R (z) < 0 by assumption (10)) is continouous, and by assumption (12) it is also monotonously decreasing. Also, notice that R (0) = 0 (as b (1), r (1), and a (0) are finite and R(0) is non-zero). Next let us prove that there is a unique level of stress where R(z) has unit elasicity, i.e. there is unique z∗ such that R (z∗ ) = − 1. Consider function ϕ(z) defined as ϕ(z) = −R(z)z. Since R(z) is strictly positive and bounded, ϕ(0) = 0 and ϕ(z) < 0 for z > 0. Furthermore according to (11) lim z→∞ ϕ(z) = 0, i.e. ϕ(z) approaches to zero from below as z tends to infinity. Hence there must be x > 0 such that ϕ (x) > 0. Take first derivative

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of ϕ(z) at z = x ϕ (x) = −R(x) − xR (x) = −R(x)(1 + R (x)) > 0. Provided that R(z) is always positive, we conclude that R (x) < − 1. Now, taking into account that R (z) is continuous and monotonically decreasing, R (0) > − 1, and R (x) < − 1 there exists a unique z∗ ∈ (0, x) such that R (z∗ ) = − 1. Notice that ϕ(z) has unique minimum at z = z∗ . Following similar lines we can prove that there is a unique z∗∗ > z∗ such that

R (z ∗∗ ) = −2. Indeed, as we have shown above ϕ(z) is a unimodal function with local (and global) minimum at z∗ . It approaches to zero from below as z tends to infinity. Thus there is x > z∗ such that ϕ (x) < 0. The second derivative of ϕ(x) at z = x is ϕ (x) = −2R (x) − xR (x) = −R (x)(2 + R (x)) < 0.

(18)

R (z) < 0

Given that the inequality is equivalent to R (x) < −2. Write down the second order condition for ϕ(z) at local minimum z = z∗ : ϕ (z ∗ ) = −R (z ∗ )(2 + R (z ∗ )) < 0. R (z∗ ) < 0,

(19) ∗  (z )

therefore R > −2. In this inequality Since R (z) is continuous and monotonous, it follows that there exist a unique z∗∗ ∈ (z∗ , x) such that R (z ∗∗ ) = −2. Now we are ready to analyze the phase diagram corresponding to dynamics defind by (4). Define function (z) as the right hand side of the Eq. (4), i.e. (z) ≡ −(c + R(z))z + S0 . Function (z) and its derivative are continuous functions of z. The zeros of (z) determine the steady states of (4) and the stability of these steady states depends on the sign of   (z). First and second derivatives of (z) are   (z) = −c − R(z)(1 + R (z))

(20)

  (z) = −R (z)(2 + R (z)).

(21)

and

Since there is unique z∗∗ : R (z ∗∗ ) = −2, (z) has unique inflection point at z = z∗∗ , and this a point of maximum for   (z), i.e.   (z ∗∗ ) = maxz   (z). Notice that if   (z∗∗ ) > 0 then (z) has two local extremes: local minimum at z1∗ (z1∗ < z ∗∗ ) and local maximum at z3∗ (z3∗ > z ∗∗ ). The phase diagram in such a case is similar to those shown at Fig. 1:   (z) > 0 for z ∈ (z1∗ , z3∗ ), and   (z) < 0 for z ∈ (0, z1∗ ) ∪ (z3∗ , ∞) If   (z∗∗ ) < 0 then (z) is monotonicaly decreasing for all z. Let us define c¯ as c¯ = −R(z)(1 + R (z ∗∗ )). Then the condition for having extremes can be written as c < c¯ . Let us assume that this condition holds. Define S1 and S2 as S1 = cz3∗ + b(a(z3∗ ))z3∗ − (n − 1)p(a(z3∗ ))z3∗ , S2 = cz1∗ + b(a(z1∗ ))z1∗ − (n − 1)p(a(z1∗ ))z1∗ . Consider (4) with S0 < S1 . Then we have (0) = S0 > 0, (z1∗ ) = S0 − S2 < S0 − S1 < 0. By continuity of (z) there is z1 ∈ (0, z1∗ ) such that (z1∗ ) = 0. This steady state is unique and stable since   (z) < 0 for z ∈ (0, z1∗ ). There are no other steady states because (z) < S0 − S1 < 0 for all z ∈ (z1∗ , ∞). This corresponds to case 1 of the proposition. Now let S0 > S2 . Notice that (a) (z3∗ ) = S0 − S1 > S0 − S2 > 0, and (b) (z) < 0 for any z > zc = S0 /c (zc corresponds to ‘no ˙ 1∗ ) = 0. Since   (z) < 0 for z > z3∗ steady state z3 is relationship’ case). By continuity of (z) there is z3 ∈ (z3∗ , zc ) such that z(z stable and unique. This is case 2 of the proposition. Finally, take S0 ∈ (S1 , S2 ). First, we have (0) = S0 > 0, (z1∗ ) = S0 − S2 < 0, and   (z) < 0 for z1 ∈ (0, z1∗ ). Thus there is a stable state z1 : (z1∗ ) = 0 and it there are no other steady states on (0, z1∗ ). Second, there is unique steady state z2 on (z1∗ , z3∗ ) because (z) is monotonically increasing, (z1∗ ) = S0 − S2 < 0, and (z3∗ ) = S0 − S1 > 0. However z2 is unstable because   (z) > 0. Third, given that (z3∗ ) = S0 − S2 > 0, (z) < 0 for z > zc , and   (z) < 0 for z > z3∗ there is unique stable steady state on (z3∗ , ∞). Case 3 of the proposition is proven. 

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