1 Introduction

The musicians wink and smile at each other as they play,

and I see then that they are the secret emissaries of a worldwide

lower-class conspiracy to snatch joy out of degradation and filth.Footnote 1

Famous musicians and composers have always had to migrate from one city to another following the courts of their patrons. Among the most famous we can cite Bach, Mozart, Haendel, or the Italian composer Lulli who changed is surname in Lully after joining the court of the king of France. This phenomenon has been studied by Borowiecki (2015) who provided an interesting economic model of cultural goods’ production of classical music composers lived between 1750 and 1899. Since, as we can see from J.S. Bach’s biography, many musicians periodically shifted from one city to another, we introduce and analyze the dynamics in the model proposed by Borowiecki (2015).

Nowadays, despite centuries have passed, the musicians’ work situation is still a peculiar one, characterized by unstable forms of employment, multi-pattern careers and many constraints and challenges for its workers (Frederickson and Rooney 1988; Barker et al. 2009; Moore 2016; Vaag et al. 2014, 2016). To persist in the music industry, musicians adopt several strategies, such as teaching, performing as a freelancer (Menger 2006) or working both inside and outside the music field. These activities, which can be ranked on various levels of prestige, budget, conditions, musical quality or role (Menger 2006), shape forms of career that in literature have been defined as ‘Portfolio career’ (Cawsey 1995), ‘Boundaryless career’ (Defillippi and Arthur 1994), ‘Protean career’ (Hall 1976), or ‘Bulimic career’ (Kong 2011). Despite the differences in their names, these careers arise from the same working situation, yet underline different aspects. The ‘Portfolio career’ (Cawsey 1995) highlights the entrepreneurial mindset necessary to carry on with the music practice, that involves for example holding different jobs and creating successful professional networks (Cawsey 1995; Bennett and Bridgstock 2015). The ‘Boundaryless career’ (Defillippi and Arthur 1994) takes its name from the practice of not being tied to any organization and focuses on motivational, profession related and networking skills, defined as ‘knowing why’, ‘knowing how’ and ‘knowing whom’ (Defillippi and Arthur 1994; Zwaan et al. 2010). The ‘Protean career’ (Hall 1976) is built upon the concept that the development of new skills and the activity of detecting job opportunities shape the musician’s identity (Hall 1976; Bennett 2009). Finally, Kong (2011) shed light on the dark side of these forms of career, defining them as ‘Bulimic career’, as they are characterized by an alternation of high working periods and breaks. In this sense, Moore (2016) claims that emphasizing entrepreneurship and portfolio careers suggests that this is the only possible answer to the precariousness in musical work, calling the musicians not only to embrace these practices, but also to endorse them. It has been in fact coined a term that reflects this practice. The word ‘flexploitation’, which comes from flexibility and exploitation, well describes the musicians’ attitude to embrace values of uncertainty and challenge, but also creativity and flexibility in their artistic careers (Morgan and Wood 2014; Schediwy et al. 2018). On one side the flexibility is a valuable feature of the musicians’ career, because it brings them the chance of pursuing innovation and diversifying the artistic product (Menger 2006). On the other, a too flexible environment may cause occupational stress (Vaag et al. 2014), self-esteem reduction, a sense of replaceability (Frederickson and Rooney 1988), emotional exhaustion, and alcohol abuse (Parker et al. 2021).

Unfortunately, sometimes, diversifying skills and building a portfolio of competencies is not enough to persist in the profession and musicians have to migrate to more cultural active cities to achieve a professional fulfillment, as their predecessors, centuries ago, did. According to Bennett (2010), artists decide to migrate to another urban center when the city they are living in no longer offers them employment opportunities, sometimes also for limited amounts of time (Hautala and Nordström 2019). However, even if artists decide to migrate, the literature showed that all over the world musicians find almost the same precarious scenario. In this sense, Guo and Wyszomirski (2019) noted that Western and Chinese musicians presented the same career characteristics. Both group, in fact, accompany the performance jobs with other occupations, such as teaching (Guo and Wyszomirski 2019). Other studies underline that countries’ policies affect the lives of professional musicians: In terms of available working opportunities for French and Switzerland (Perrenoud and Bataille 2017), number of freelance workers for the United Kingdom and Germany (Harper 2001), and forms of unreported employment in Italy (Balestrino 2012). Thus, the differences between cities and countries are an important issue for musicians, as they can be translated in available working and learning opportunities. Finally, several scholars investigated how and why musicians continue in persisting in musical career even with its related insecurity and potential negative outcomes. The answer resides in their motivation, as noted by Chafe and Kaida (2020). Motivation may act both as a driver and as a protection factor against the difficulties (Vaag et al. 2014; Dobrow Riza and Heller 2015). Hence, when considering the musicians’ scenario it is important to bear in mind that these workers are driven not only by economical incentives, but also by passion and calling. As defined by Freidson (1990), the music practice is a ‘labor of love’.

Changing the perspective to adopt that of cities, it is important to remember, following (Florida 2002; Ercolano et al. 2017; Li et al. 2018) that cities benefit from the organization of cultural events, not only in economic terms, but also in terms of image. Richards and Wilson (2004) found three aspect related to this phenomenon: The ‘halo effect’ (Hall 1992), ‘feel-good effect’ (Allen et al. 2002), and the ‘showcase effect’ (Fredline and Faulkner 1998), which are mechanisms that transfer the aspects of a cultural event to the city that hosted it. Cerisola and Panzera (2022), as reported in Sect. 2, further observed that favouring creative activities fosters the economic and geographical development of a city with pulling effects. To this extent, we can argue that a cultural city could not only be a benefit for the artists that live in it, but also for the community itself. Starting from these considerations, the goal of this paper is to present a dynamical model of musicians’ career choices and migrations adopting Borowiecki (2015)’s analysis of musicians’ crowding. Following Borowiecki (2015), for the sake of simplicity, we consider a single city and study the dynamics of artists’ migration choices. We can see how the dynamic aspects follow naturally from Borowiecki (2015) and how they can be useful to interpret this phenomenon. In fact, as musicians choose between staying in the city and migrate and their choice affects the others in terms of crowding costs (Borowiecki 2015), this is one of the “either-or situations, not choices of degree or quantity” studied in (Schelling 1973, p.381).

The paper is organized as follows. In Sect. 2 we discuss the musicians’ working scenario incorporating the contributions from agglomeration economies, musicians’ agglomeration, dynamical analysis, sustainable development indicators in the model by Borowiecki (2015). In Sect. 3 we provide the formalization of the dynamical model and in Sect. 4 its analysis. Section 5 provides the interpretation of the results in terms of career’s sustainability choices. Finally, Sect. 6 is dedicated to future research directions and theoretical and practical implications.

2 Theory

The purpose of this section is to describe the theoretical framework and the methodology adopted to describe the musicians’ mobility choices.

Migration and work mobility are complex phenomena with tangible repercussions on people’s wellbeing and national economies. In the last years, institutions have stressed the importance of analyizing the theme of migration underlying its link with the policies of sustainable development (e.g., UN General Assembly 2015). This encouraged the development of indexes that rate states’ policies and promote best practices (see Solano and Huddleston 2020). One of the most adopted and reliable index in terms of integration policies is the Migrant Integration Policy Index 2020 (Solano and Huddleston 2020)—from now on MIPEX 2020—which contains 8 policy areas and considers 56 countries. The eight areas considered by the MIPEX 2020 as important to assess the countries’ policies are: labour market mobility, family reunion, education, health, political participation, permanent residence, access to nationality, anti-discrimination (Solano and Huddleston 2020). The present study intends to investigate and shed light to the first of these indicators: the labour market mobility. According to MIPEX 2020, the most important factor in finding and obtaining employment consists of the skills held by the worker (immigrant or not) in a particular social context and historical moment (Solano and Huddleston 2020). However, possessing the right skills is not a sufficient condition for the development of a sustainable career. In fact, according to De Vos et al. (2020), three dimensions must be considered when studying the sustainability of a career: person, context and time. The analysis of the person involves, for example, physical and psychological well-being, personal characteristics and motivations. Context analysis investigates the elements and resources available, from a systemic perspective, around the individual. Contextual elements refer to the ties between people (e.g., the network of colleagues), the political and social situation and the presence of available economic resources. Finally, the temporal dimension of the theoretical framework is crucial to understand whether the characteristics of the career undertaken by the individual are sustainable not only in the immediate term, but also over time. As we have seen with MIPEX 2020, not only academic environments are addressing the topic of sustainability, but also international institutions (Solano and Huddleston 2020). Systemic sustainability of individuals is in fact a recurring theme in the Sustainability Goals promoted by the United Nations as well (UN General Assembly 2015). In four of the seventeen goals placed sustainable development of cities (goal 11, target 11.4), careers (goal 8, target 8.3), education (goal 4, target 4.7) and mobility (goal 10, target 10.7) play a key role. On the same page, we find the definition provided by the United Nations Sustainable Goals Development for 2030 for international migration: “international migration is a multidimensional reality of major relevance for the development of countries of origin, transit and destination” (UN General Assembly 2015, p. 8). Following this strand of studies and indicators, we consider here the effects of congestion deriving from musicians’ agglomeration and mobility and analyze the phenomenon in its multidimensionality under the framework of agglomeration economies (Duranton and Puga 2004; Borowiecki 2015). To do so, we adopt a dynamical model to better analyze the development of the phenomenon over time, by taking into account its multiple and mutable facets.

As discussed by Glaeser (2010), “agglomeration economies are the benefits that come when firms and people locate near one another together in cities and industrial clusters” (p. 1). Duranton and Puga (2004) finds three mechanisms at the bottom of agglomeration economies: sharing, matching and learning. If we go deeper into the subject, as already noted by Borowiecki (2015), we can find many similarities and points of connection between the behavior of musicians and agglomeration economies.

The first mechanism, sharing, manifests itself in people who shares indivisible gains, facilities and risks (Duranton and Puga 2004). Musicians tend to agglomerate in cities that present a finite number of facilities, such as concert halls, music schools or festivals (Duranton and Puga 2004; Grant and Buckwold 2013; Borowiecki 2015). By agglomerating in a same city, musicians also benefit from Risk sharing, which is related to labour pooling (Duranton and Puga 2004). On one hand workers benefit from agglomerating because it reduces the risk of unemployment, on the other firms benefit from workers’ agglomeration because it allows them to have an adequate number of available resources (Duranton and Puga 2004).

The second mechanism, matching, refers to the benefits that come from matched skills which result in better outcomes (Helsley 1990; Duranton and Puga 2004). In fact, increasing number of musicians and artistic related occupations, enable workers to find job opportunities more related with their skills. In Europe the most music active cities are also the ones that present different opportunities for musicians, in terms of festivals, opera houses, or academies (Montalto et al. 2019). Musicians have to frequently fall back onto teaching, when no other working opportunities are available or do not fully allow them to make a living. Hence, it would be of mutual gain a situation in which there is a perfect match between the demand and the offer of skills.

Finally, large cities, bringing together a considerate amount of people, present many learning opportunities, that is the third mechanism, learning (Duranton and Puga 2004). This is particularly evident for musicians, as already underlined by Borowiecki (2015), when he found that the composers’ crowding was associated to artistic production. On the other hand, it is well known in literature (Bartleet et al. 2020) and well displayed by international educational programs (European Commission and Directorate 2018), that for a young musician an abroad experience in an artistic vibrant city is crucial for a successful career development. Furthermore, a recent study by Cerisola and Panzera (2022) showed that promoting creative activities and occupations enhance the city’s economic growth, by fostering the creation of new skills, the development of prosocial behavior and resilience. Finally, following Duranton and Puga (2004), the musicians’ crowding can act as a signaling device (Spence 1973).

The phenomenon here briefly presented of musicians’ working mobility is multi-facets and complex with relevant implications on the social environment and people’s lives (Shelley and Wagner 1998). Therefore, we analyze it under the framework of “Chaos Theory" and “Complex Adaptive Systems Theories" (see paragraph 3.1) to provide a multidimensional overview of the topic. Taking into account the peculiarities of the musicians’ labor market and the social context, we develope the following research questions to analyze how the phenomenon of musicians’ mobility develops in time and what are its consequences on the social contest.

  1. 1.

    First, we investigate how sustainable mobility choices develop over the course of musical careers and how these are influenced by the social context as described in De Vos et al. (2020).

  2. 2.

    In this regard, secondly, we consider the role of externalities in the musicians’ group and study their influence on choices as suggested in Schelling (1973).

  3. 3.

    Finally, we are interested in analyzing whether the assumptions and considerations of Borowiecki (2015) also hold up in a dynamic analysis.

3 The dynamic model

3.1 Methodology and analysis strategy

The contributes mentioned in Sect. 2 and the research questions will be analyzed under the paradigm of the “Chaos Theory”. According to the Dictionary of the American Psychological Association (APA) chaos theory is “an area of mathematical theory that deals with nonlinear systems that are profoundly affected by their initial conditions, tiny variations in which can produce complex, unpredictable, and erratic effects” (American Psychological Association 2023). Chaos Theory is a useful approach to study phenomena that at first appear inexplicable, but actually conceal recursive patterns (Shelley and Wagner 1998), as the multidimensionality of musicians’ mobility. Furthermore, consistently to Complex Adaptive Systems Theories, we considered musicians behaving as agents that seek to maximize their profits. That is, we postulate that musicians are rational and behave in their best interest, e.g., they leave their city they are in if the city no longer offers them working opportunities. In particular we adopt a dynamical model that considers binary choice with externalities (Schelling 1973). Binary choices with externalities have been widely studied by Schelling (1973) and other scholars (e.g., Dal Forno and Merlone 2019), and refer to the choices between two alternatives individuals make taking into account also their believes about other people’s choices. The choice musicians are called to make is whether to stay in the city or to leave it. This choice is influenced by different parameters that reflects the characteristics of the music job market and the ones of the city. By formalizing those parameters and introducing dynamics, it is possible to draw a model of musicians choices that varies along time. This has already been done for other studies (e.g., Bischi et al. 2009) and the use of dynamical models for interpreting phenomena is largely used in various scientific sectors (Sushko et al. 2016).

Section 3.2 illustrates the model and Sect. 4 provides its analysis and comparison with real data.

To conduct the analysis we used: Excel and C++. The map of the dynamic model has been first written in C++ and then controlled using Excel as in Merlone et al. (2008). The figures have been made using Gnuplot and R. The musicians’ data to support the model have been gathered via Statista, proper references are included in the text.

3.2 The model

Many different scientific fields adopt dynamical systems to analyse their phenomena, such as psychology (Rinaldi and Della Rossa 2019), sociology (Shelley and Wagner 1998), economics (Brianzoni et al. 2015), mechanics (Brogliato 1999), and electronics (Avrutin et al. 2015; Sushko et al. 2016). Hence, starting from the considerations made above based on musicians and agglomeration economies’ literature, we propose a model of musicians’ migration which expand the one presented by Borowiecki (2015). In his paper Borowiecki (2015) aimed to investigate the artistic productivity of classical music composers lived between 1750 and 1899. He did that by building a theoretical model which reflected the benefits that come from agglomeration economies and their disadvantages, the crowding costs. He further corroborated his model by using the data on composers’ birthplace and famous compositions, that revealed consistency with his theoretical formalization.

As in Borowiecki (2015) we consider a city with a normalized population in which \(x_t\) is the fraction of artists at time t. We assume that the amount of cultural goods depending on the fraction of artists is a sigmoid function (Cybenko 1989), which, with its S-shaped curve, describes systems with a carrying capacity determined by limited resources (Sterman 2000), as the qualitative graphical representation provided in Borowiecki (2015, p.448) suggests. A good example to replicate the shape of the curve proposed in Borowiecki (2015) is:

$$\begin{aligned} F\left( x\right) =-2 x^3 + 3 x^2 \end{aligned}$$
(1)

which is monotonically increasing on \(\left[ 0,1 \right]\) and has an inflection point in \(x=1/2\).

As in Borowiecki (2015), we assume that the considered city has a preference for cultural goods \(\gamma\) and that the agglomeration of musicians in a same city is associated with a crowding cost \(C\left( x_t\right)\). Therefore, the level of cultural output \(\Omega\) at time t depends on the fraction of artists \(x_t\):

$$\begin{aligned} \Omega \left( x_t\right) =\gamma F\left( x_t\right) - C\left( x_t\right) . \end{aligned}$$
(2)

The city’s appreciation of cultural goods, \(\gamma\), is an important parameter in the analysis of musicians’ choices as we will see in Sect. 4.

To this extent, the European Commission developed a Monitor of Cultural and Creative cities that includes 29 indicators, 9 dimensions, and 3 facets. The latters refer to the Cultural vibrancy, Creative economy, and Enabling environment. European cities are ranked according to this index which offers insights on how much the city is interested in the consumption of cultural goods (Montalto et al. 2019). This monitor helps providing insights on the city’s artistic vibrancy and offers empirical data useful for this model (see Sect. 4.1). Furthermore, we assume that the density of artists in the society, that is the amount of musicians in the city, depends on \(\Omega\), the cultural output:

$$\begin{aligned} x_{t+1}=g\left( \Omega (x_t)\right) . \end{aligned}$$
(3)

Putting together we obtain the following dynamics:

$$\begin{aligned} \left\{ \begin{array}{l} \Omega (x_t)=\gamma F\left( x_t \right) - C\left( x_t\right) \\ x_{t+1}=g\left( \Omega (x_t)\right) . \end{array} \right. \end{aligned}$$
(4)

Hence, when pursuing artistic career in their city, agents’ utility depends on the cultural goods produced. By contrast, the utility \(\overline{U}\) of agents when they do not stay in the city is considered exogenous, that is a variable which cannot be controlled in the model; for a discussion about endogenous and exogenous variables in modelling see Sterman (2000, p. 94-96). Additionally, as migration is an either-or choice (Schelling 1973), the musicians’ mobility choice is: Staying in the city vs not staying in the city, with switching propensity \(0 \le \alpha \le 1\). Therefore, putting together we obtain:

$$\begin{aligned} x_{t+1}=f\left( x_t\right) = \left\{ \begin{array}{lcl} x_t-\alpha x_t=\left( 1-\alpha \right) x_t &{} \text {if} &{} \gamma F\left( x_t \right) - C\left( x_t\right) < \overline{U}\\ x_t &{} \text {if} &{} \gamma F\left( x_t \right) - C\left( x_t\right) = \overline{U}\\ x_t+\alpha \left( 1-x_t\right) =\left( 1-\alpha \right) x_t+\alpha &{} \text {if} &{} \gamma F\left( x_t \right) - C\left( x_t\right) > \overline{U}.\\ \end{array} \right. \end{aligned}$$
(5)

where \(f = g \cdot \Omega\).

Furthermore, when \(\alpha =0\) the map (5) is the identity; when \(\alpha =1\) the map is trivial: Therefore we will consider \(0<\alpha <1\). Map (5) shows three different scenarios with respect to \(\overline{U}\) that will be analyzed in the following section (See Sect. 4).

4 Analysis of the dynamics

The mathematical properties of a map similar to (5) have been studied in Bischi et al. (2009). In this paper we focus only on the theoretical implications of the model; the mathematical analysis is available from the authors upon request. Starting from the Map (5), we can delineate three possible cases at varying levels \(\gamma\), i.e. the society’s preference for cultural goods. The larger the \(\gamma\) value, the greater the society’s appreciation of cultural goods.

Fig. 1
figure 1

The role of \(\gamma\) and \(\overline{U}\) in (5), with \(\gamma = 1.2 > 1\) which represents a city rather interested in cultural goods. \({\overline{\Omega }}\) is the maximum net benefit, \({\underline{\Omega }}\) is the all-or-nothing threshold, \(\overline{x}\) is the population that maximizes the cultural output

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Fig. 2
figure 2

Dynamical representations relative to Fig. 1, with \(\gamma = 1.2 > 1\) and \(\alpha = 0.45\). The x axis represents the density of musicians at time t and the y axis represents their density at time \(t+1\) depending on \(\overline{U}\) and \(\gamma\). The discontinuity points \(d_1\) and \(d_2\) appear where the density function of artists intersects the reservation utility level \(\overline{U}\), denoting a change in their mobility choices. In (a), there is only one discontinuity point \(d_1\), denoting the case where either every musician leaves the city or everyone stays depending on the initial condition. (b) represents the case when the musicians either follow cycles or leave the city (again depending on the initial condition), that is when there are two discontinuity points \(d_1\) and \(d_2\). In (c), since the reservation utility level does not intersect the density function, there are no discontinuity points; therefore, all musicians leave the city to pursue their music career

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In Fig. 1 we depict the cultural output as a function of musicians’ density. With \(\gamma >1\), that is a city rather interested in musical activities, there are three possible outcomes depending on the musicians’ density. If \(x < d_1\), that is a low artists’ density, the musicians in the city are not satisfied with their current situation; if \(x > d_1\), musicians will come back to the city. In Fig. 1a, in fact, \(\overline{U}\), i.e. the musicians’ reservation utility level, is between the all-or-nothing threshold \({\underline{\Omega }}\) and 0; we will see in the next paragraph why \({\underline{\Omega }}\) is called the all-or-nothing threshold. Figure 1b shows the situation where the reservation utility level \(\overline{U}\) is between the all or nothing threshold \({\underline{\Omega }}\) and the maximum net benefit \({\overline{\Omega }}\), that is the maximum benefit musicians can obtain from pursuing their occupation. Here, if \(x < d_1\) or \(x > d_2\) the musicians living in the city are unsatisfied, while if \(d_1< x < d_2\) the musicians outside the city are unsatisfied. Finally, when the reservation utility level is higher than the maximum net benefit \({\overline{\Omega }}\) we can see that, as in 1c, even if there is a great appreciation of cultural goods, the musicians living in that city are unsatisfied, whatever the density is. Eventually, we can expect that all the musicians will leave the city. On the other hand, the case of a city with less interest in cultural goods, that is \(\gamma \le 1\), in terms of dynamics boils down to the scenario of Fig. 1b and c.

In summary, if the density is greater than \(d_1\), the musicians stay in the city or come back to the city. If the musicians’ density is lower than \(d_1\) a fraction of artists is unsatisfied with its current situation. Finally, if the density is equal to \(d_1\) or \(d_2\), that is when the density is such that the utility when pursuing an artistic career is the same as the reservation utility, musicians are indifferent: They both can stay in the city or leave it.

Furthermore, in Figs. 1 and 2 we can notice the presence of intersection and discontinuity points, which are respectively denoted as \(d_1\) and \(d_2\). Without diving into the mathematical details, discontinuity points can been seen as points where the function exhibits a “surprising” behavior (for instance in Fig. 2a we can see a “jump” in the function) and are an important feature in dynamical modelling (see, Sushko et al. 2016). Therefore, from (5) we obtain:

$$\begin{aligned} x_{t+1}= \left\{ \begin{array}{lcl} x_t-\alpha x_t &{} \text {if} &{} x_t<d_1 \vee x_t>d_2 \\ x_t &{} \text {if} &{} x_t=d_1 \vee x_t=d_2\\ x_t+\alpha \left( 1-x_t\right) &{} \text {if} &{} x_t<d_1 < d_2.\\ \end{array} \right. \end{aligned}$$
(6)

In Fig. 1, if we focus on the intersection points \(d_1\) and \(d_2\), when assuming that \(d_1 < d_2\) are distinct and inside \(\left[ 0,1\right]\), we can represent the musicians’ dynamics as in Fig. 2 and see that \(d_1\) and \(d_2\) are discontinuity points. Figure 2 quantifies the densities in the next round, by using Equation (6) to determine the density of artists at time \(t+1\), where \(\alpha\) is the fraction of unsatisfied musicians that denotes its migration propensity. As a matter of fact, Fig. 2a–c are respectively the dynamical representation of Fig. 1a–c, as they show the musicians’ mobility choices at time \(t+1\). In Fig. 2a, when the reservation utility level \(\overline{U}\) intersects the density function of artists, we have the discontinuity point \(d_1\) that denotes a change in the musicians’ mobility choices. From that amount of musicians, every artist pursue the music occupation in the city. In other words, in the scenario of Fig. 2a, either every musician leaves the city or everyone stays. In fact, as we have seen in the previous paragraph, the reservation utility level for those artists is between 0 and \({\underline{\Omega }}\), as the words suggest, the all-or-nothing threshold. Figure 2b represents the scenario where cycles of different periodicity happen (we will properly discuss this phenomenon in the next paragraphs). Since the reservation utility level \(\overline{U}\) intersects the density function in two points, the two discontinuity points \(d_1\) and \(d_2\) appear. Between these discontinuity points, the musicians’ behavior follow cycles, as they periodically leave or stay in the city. As we have seen above with Fig. 1b, we have periodic cycles when the reservation utility level \(\overline{U}\) is between the all-or-nothing threshold \({\underline{\Omega }}\) and the maximum net benefit \({\overline{\Omega }}\). Finally, Fig. 2c is the dynamical representation of 1c. Since the reservation utility level \(\overline{U}\) does not intersect the density function of artists, that is \({\overline{\Omega }}<\overline{U}\), there are no discontinuity points. This represents in fact the scenario where musicians have to leave the city to pursue their professional occupation. We have seen that when the reservation utility level \(\overline{U}\) intersects the density function, the two discontinuity points \(d_1\) and \(d_2\) appear, denoting the occurrence of period cycles (see Fig. 2b). In this case, the state of the system repeats cyclically and the number of time interval necessary for recurrence is called “period”. In other words, from Fig. 2 we can notice that when the blue line is above the bisector musicians come to the city, when the blue line is below the bisector musicians leave the city.

To summarize, the reservation utility level \(\overline{U}\) plays a major role in determining the possible scenarios. If the reservation utility level (\(\overline{U}\)) is higher than the maximum net benefit \({\overline{\Omega }}\), every musician has to leave the city. If \(\overline{U}\) is between the all-or-nothing threshold (\({\underline{\Omega }}\)) and the maximum net benefit (\({\overline{\Omega }}\)), only a portion of artists stays in the city. Finally, if the reservation utility level (\(\overline{U}\)) is between 0 and the all-or-nothing threshold (\({\underline{\Omega }}\)), and \(\gamma\), i.e. the city’s appreciation of cultural goods, is higher than 1, we have an either-or situation: Either every musician stays in the city or everyone leaves.

Fig. 3
figure 3

Basins and periodic cycles of map 5 when \(\gamma =0.50\) and \(\overline{U}=0.017\). With respect to \(\alpha\), for instance, we can easily observe the presence of 2-cycle, 7-cycle, 12- cycle, 5-cycle

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In order to better understand the phenomenon of musicians’ mobility it is possible to analyze the different cycles and their periodicity. Figure 3, for instance, shows that at varying levels of \(\alpha\), i.e. the musicians’ propensity to switch choice, there are different possibilities and configurations, with respect to the values of \(\gamma\) and \(\overline{U}\). In other words, different period cycles happen depending on the musician’s propensity to switch choice. If we set \(\gamma =0.50\) and \(\overline{U}=0.017\) we can observe the situation depicted in Fig. 3. On the x axis there is the propensity \(\alpha\) to switch choice. On the y axis there is the fraction of musicians at time t. As \(\alpha\) increases, cycles of different periodicity happen. For example, let us consider a 2-cycle (red in Fig. 3). With \(\alpha \simeq 0.4\) musicians periodically leave or stay in the city, depending for instance, on their careers’ goals. The other attractor, with \(\alpha \simeq 0.4\) and depending on the value of \(x_0\), is the basin of the origin (black in Fig. 3). In general, the presence of the basin of the origin means that whatever the values of \(\alpha\), \(\gamma\), or \(\overline{U}\) are, the origin always represents a stable and fixed point. In musicians’ terms it implicates that leaving the city is always a valid choice in all of the possible scenarios, as documented in the literature. We remind the reader that the basins of attractions are the set of initial conditions whose trajectoriesFootnote 2, in this case, either go to the origin (every musician leaves the city) or to a cycle (Nusse et al. 1994). Finally, Fig. 3 presents a peculiarity, between the \(\alpha ^*\) and the beginning of the 2-period cycle there is a “black stripe” that represents the basin of the origin. When musically interpreting this scenario, one could mistakenly infere that every musician decides to leave the city, yet when \(\alpha\) increases, that is an increasing propensity to switch choice, the 2-cycle reappears. This shows that, even in the stylised model we presented, the musicians’ working situation is rather complex and mutable and many factors have to be considered to give appropriate interpretation.

4.1 An empirical example: the case of musicians in the United Kingdom

Fig. 4
figure 4

Estimated number of musicians in the United Kingdom from 2010 to 2021. Source: Office for National Statistics (2021)

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The presence of cycles in the number of musicians employed in one city is not just a result of the model, yet it can be empirically observed in the reality. We decided to take into considerations the case of the United Kingdom. As Cottrell (2002) states, the capital of the United Kingdom, London, is one of the most vibrant musical city in the European scenario, in terms of musicians employed, facilities and musical genres. If we extend our reasoning to the whole country, in Fig. 4 we can observe the behavior of musicians employed in the United Kingdom from 2010 to 2021. First, we can notice an increasing trend in the amount of artists employed, secondly it seems that the years can be divided into groups of four, as they per se constitute a pattern that persist over time: From 2010 to 2013, from 2014 to 2017, and from 2018 to 2021 (Office for National Statistics 2021).

Hence, to prove the consistency of our model we provide here a formalization. In Fig. 5 we can see the dynamical representation of musicians’ behavior in the UK. In the previous section we have seen that cycles of different periodicity happen with respect to the values of \(\alpha\), \(\overline{U}\), and \(\gamma\). Hence, it is possible to find a constellation of parameter values resulting in a qualitatively similar cycle 4. When analyzing the parameter values it is interesting to notice that \(\gamma \simeq 0.95992\) denotes a relatively high appreciation of cultural goods compatible with the situation in the UK as reported by the European Monitor of Cultural and Creative Cities (Montalto et al. 2019). On the other hand, following Smith and McBride (2021), the reservation utility level \(\overline{U}\) is relatively low: \(\overline{U}=0.07\). Therefore, in Fig. 5 we can see represented the Map (5) producing a 4-cycle, that resembles the behavior of Fig. 4. This formalization shows that even the simple model we presented may be consistent with real data.

Fig. 5
figure 5

Dynamical formalization of musicians’ behavior in the United Kingdom. Map (5): 4-cycle; with \(\gamma \simeq 0.9599\), \(\overline{U}=0.07\), \(\alpha \simeq 0.6553\)

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5 Discussion

In the previous paragraphs it has been described a model which aims to illustrate the mobility choices a musician makes during her career. Starting from Borowiecki (2015) and Bischi et al. (2009)’s formalization of binary choice dynamics, several factors that may influence a career choice have been delineated: the city’s appreciation of cultural goods, the reservation utility level and the propensity to switch choice. All these aspects have been further influenced by the presence of other people making the same choices at the same time and one own’s prediction of the decisions of others. This produced a dynamical model of binary choices with externalities (Schelling 1973).

The first aspect analyzed is the effect of \(\gamma\), the city’s appreciation of cultural goods. The parameter \(\gamma\) is a core element in the model and it is largely supported by the literature on musicians’ and creative mobility. In fact, as Lee (1966) states, a city that does not offer many opportunities or is geographically isolated from other urban centers may act as a ‘push factor’ for musicians to migrate from (Bennett 2010). On the other hand, a creative city or a city with already a large pool of musicians can be attractive to migrate to (Borowiecki 2015). As we can see from Fig. 2c, when the reservation utility level (\(\overline{U}\)) is large enough, every musician has to leave the city, despite the city’s appreciation of cultural goods. This is rather interesting, if we further observe that, even when musicians are in an active and creative city, the choice to migrate is always a stable fixed point, as we can see from Fig. 2b and c. These findings are not surprising if we consider the literature on creative migrations, music education and creative programs. In fact, as we mentioned before, the European Commission and Directorate (2018) fosters the mobility between countries for young musicians. Additionally, for more established artists, migrating constitutes an opportunity to broaden the portfolio of competencies and the network, gain visibility, and find more self-fulfilling job opportunities (Lee 1966; Bennett 2010).

A second important factor to consider is the propensity to migrate \(\alpha\), which influences the decision. The parameter \(\alpha\), in fact, may vary in reason of family ties or social connections a musicians might have in the original city (Bennett 2010). Similarly, the parameter \(\alpha\) can be influenced by the integration policies of the destination city. In fact, according to MIPEX 2020, only in 8 of the 56 nations analyzed there were equal job guarantees for immigrants as for residents. Notably, the most virtuous states in this regard are those in Western Europe and Canada (Solano and Huddleston 2020). The present model clearly reflects this aspect. As we have seen in Fig. 3 a different propensity to stay influences the development of diverse period cycles, namely the times a musician leaves and returns to her city.

In general, as we can see from Fig. 1a, the number of musicians needs a subside, as the literature on agglomeration economies and Borowiecki (2015) already noted, and the stability in a same city is not fixed, even when the city’s preference of cultural goods is large and the musicians’ migration follows cycles. We can observe this behavior if we look at the data on musicians’ employment in the UK (Fig. 4). As Fig. 4 suggests, there is not a constant and equal number of musicians over the years, yet it varies. The subside of artists also suggest a well-known phenomenon in music employment literature. In order to proficiently pursue the artistic occupations, musicians need a network of peers to rely on. This happens mainly for two reasons: The first is related to creating a network for working opportunities, such gigs or pupils; the second, relates to the need of psychological support that a profession with such high mental demands requires (Zwaan et al. 2010).

Finally, the last possible scenario, where the city’s appreciation of cultural goods allows every musician to stay in the city, is theoretical. One possible exception that could confirm the consistency of Fig. 2a is the city of Nashville (Harper et al. 2012). However, this is inconsistent with the literature on the phenomenon. In fact, even if the musicians’ working environment is characterized by instability, moving to another city often offers to the musicians more learning, working and network developing opportunities. Nonetheless, the migration of musicians to and from a city contributes to the city’s development (Cerisola and Panzera 2022), potentially enriching the musical products.

6 Conclusions

The musician working scenario is characterized by many forms of instability that on one hand offer flexibility and autonomy and on the other economic and psychophysical problems. As we have seen, the hindrances to the occupational stability could bring stress, emotional exhaustion, alcohol abuse and other psychophysical pathologies. However, hundreds of years of music practice provided many resources and insights to the musicians to strive (e.g., Borowiecki and Kavetsos 2015; Borowiecki and Møller Dahl 2021). The first is relying and widening the social network to have access to more working opportunities, getting new skills and gaining social support, a valid ally for the psychological challenges brought by the profession. The second, well recognized in the recent literature, is an entrepreneurial mindset, which consists in professionals that actively promote their work and constantly seek for new job opportunities. Hence, the aim of this paper was to provide a dynamical model of musicians’ career choices taking into account different aspects of their peculiar working scenario, focusing on the conditions that make musicians consider moving to another city and considering the role of other musicians in shaping the decision of staying or leaving the city.

In summary, the model showed that, even if the city’s appreciation of cultural goods is a strong factor in predicting mobility choices, the phenomenon of migration is an integrant part of the musicians’ career and its sustainability. In fact, we have seen in the model that leaving the city is always a stable fixed point, both to enrich the musical skills and to seek new job opportunities that may contribute to the career and personal development (De Vos et al. 2020). Secondly, the model also illustrated that the density of musicians in a city may also contribute in determining the musicians’ mobility choices, which are always influenced, under the assumption of agents’ rationality, by the reservation utility level and choices’ externalities (Schelling 1973). Thirdly, even though Borowiecki (2015) considered classical music composers born between 1750 and 1899, his findings on the optimal number of active composers in the same city are consistent with the dynamical model. Furthermore, our model (as well as Borowiecki (2015)’s one), is compatible with the literature on agglomeration economies, since it shows that the density of musicians concurs in determining mobility choices and, by extent, the density of musical communities present in the cities. Finally, Paragraph 4.1 presented a first attempt on how to combine dynamic models and empirical data. The model proposes, in fact, a theoretical formalization that could easily serve as a base for studying migration propensities, as we have seen with the case of musicians in the United Kingdom. Since the phenomenon of migration and work mobility is a notable one, such dynamic models, combined with empirical indicators, as the Cultural and Creative Cities Monitor by the European Union (Montalto et al. 2019), the MIPEX 2020 (Solano and Huddleston 2020) or the Sustainable Development Goals Indicators (UN General Assembly 2015), could help bringing insights on the effects of the guidelines on migration policies and migrants’ sustainable integration, as it has been already done in other fields, such as, for instance, with management flight simulators (Elsawah et al. 2017).

6.1 Limitations and future research directions

For future studies it could be useful to adopt a quadratic function instead of the cubic one here adopted. A sigmoidal function (Udell and Boyd 2015), in fact, could help addressing many other research questions. For instance, in the light of the recent pandemic, it would be interesting to examine the impact that a worldwide event can have on musicians’ career choices, especially in terms of geographical mobility. The fostering of remote working and digitalization practices might have influenced the careers’ perspectives and brought new features in the musicians’ employment. In fact, as we can see in Fig. 4, the number of active musicians is lower in the UK after the pandemic. A phenomenon that could also be possibly related to the increased job insecurity among musicians after the pandemic and the BREXIT’s consequences (Montalto et al. 2021; Alfarone and Merlone 2022; Rozbicka et al. 2022). Additionally, the recent war between Russia and Ukraine has, once again, meant additionally changes in the international mobility of artists (Hall 2022). Furthermore, in further research it will be interesting to include several communities with different policies, consider relocation costs and also the choice of leaving the musical career as studied in Merlone and Alfarone (2023). Such a complex model, will not entail binary choices and several other mathematical aspects will need to be considered. Another possible area of research refers to gender differences in the musicians’ entrepreneurial practice and their effects on different migration propensities. The literature, in fact, shows that men and women differs in their entrepreneurial activity. According to Miller (2016), the entrepreneurial artistic practice requires activities that are more socially acceptable if performed by men, such as self-promoting, taking risks, asking for help and building social networks. Furthermore, the literature on creative migrations, reports that family ties could influence the mobility perspectives (Bennett 2010). The final issue regards considering more cities and their related different propensities to migrate to. In fact, different cities might be more or less artistically attractive and this could varies from musical genre to musical genre, as we can see from the Monitor of Cultural and Creative cities (Montalto et al. 2019). Hence for future studies, the purpose is to adopt a different function and to model musicians’ mobility choices with respect to the global events. Finally, the practicality of a dynamic model can, as we have seen, be useful in analyzing and describing complex phenomena such as international labor mobility. In a future development of the model, it also might be interesting to give more space to indicators such as MIPEX 2020 (Solano and Huddleston 2020) or the Sustainable Development Goals (UN General Assembly 2015) in order to understand which of these might play a more fundamental role in sustainable career development in music.

6.2 Theoretical and Practical implications

When we think about music, several aspect may come to mind: a song, a musician, different music genres, an orchestra, a band, or a music tour. Rarely, we refer to the complex employment scenario and the non-canonic forms of career musicians undertake during their professional lives. Nevertheless, the implications of these aspects directly reflect themselves on the economic wealth of a city. In particular, promoting creative activities and creative occupations seems to foster the economic growth of an urban center (Cerisola and Panzera 2022). At the same time, for musicians is vital to engage in activities in new places and forming new networks. The model, starting from the literature and taking into account its evidences, showed a similar pattern. Therefore, as the European Union is already doing (European Commission and Directorate 2018), it is important to foster in music students a proactive mindset about geographical mobility, in terms of abroad times during the studies. Bartleet et al. (2019), in fact, list alongside the important matters to address in the higher music education curricula, such as entrepreneurship, gender parity, digitalization and psychophysical well-being, also mobility as a possible solution to the diminishing employment opportunities in Australia.

In conclusion, the music job market is a peculiar one that well suits for theoretical investigations and presents many challenges in its analysis and formalization. The model here proposed represents an attempt to describe the migration choices, taking into account many important different aspects related to the matter. As the situation is rather trivial to study, it is hoped that this model could act as a useful tool to further studying the career features of migrating musicians and other workers’ migration behavior.