The issue of immigrants in Italy: a rational model of immigration management by Italian municipalities

Consider a single Italian municipal authority whose utility U _R (ψ(t)) at time t is related to the revenues R(t) (>0)⁷ provided by the central government for the flow of immigrants, ψ(t) (≥0), which can be relocated to the city hosting facilities⁸. Specifically, the revenues of the municipal authority deriving from the accommodation of immigrants, \(R(\psi): \mathbb {R_{+}} \mapsto \mathbb {R{}_{}}\), increase in the number of immigrants entering its jurisdiction⁹.

U _R (ψ(t)) is assumed to satisfy the Inada conditions (i.e., being positive, strictly monotonically increasing, and concave, that is U _R (ψ(t))>0, U ^′ _R (ψ(t))≡U ^′ _R (·)>0 and U ^′′ _R (ψ(t))≡U ^′′ _R (·)<0, with \({\lim }_{t \to +\infty }U^{\prime }{{~}_{R}}(\psi (t)) = 0\phantom {\dot {i}\!}\) and \(\phantom {\dot {i}\!}{\lim }_{t \to 0}U^{\prime }{{~}_{R}}(\psi (t)) = +\infty \)).

The hosting facilities have an optimal level of absorption, which coincides with the maximum stock of immigrants which can live in the facility, \(\bar {\pi }(t)\). Nonetheless, assume that in order to reduce congestion, the authority can arbitrary utilize, at any time t, a share of the revenues to partially increase the optimal level of absorption of the facility, so that the latter is not static but can change over time.

where ξ (>0) represents the decongestion parameter denoting the rate at which immigrants are expelled from the municipality. Suppose then that deviations from the optimal congestion level provoked by the flow of immigrants are detrimental to the municipal authority according to the following functional form: \(\Omega [\!\pi (t) - \bar {\pi }(t)]\); this disutility cost faced by the authority is assumed to be increasing and convex \(\left (\mathrm {i.e.}, \Omega '(\cdot) \equiv \frac {\partial \Omega [\cdot ]}{\partial \pi (t)}>0 \text {~and~} \Omega ^{\prime \prime }(\cdot) \equiv \frac {\partial ^{2}\Omega [\cdot ]}{\partial \pi (t)^{2}}>0\right)\). In other words, this disutility cost of congestion will be considered by the municipal authority when setting the quotas of immigrants to be allowed in the city. With a total number of immigrants close to zero, the city will have no congestion in its hosting facilities, but the authority’s utility will be null, since revenues from immigration will be zero. On the other hand, when the total number of immigrants increases, the municipal authority faces an additional trade-off between higher revenues together with higher congestion costs (if none or a reduced share of the revenues is utilized to decongestionate the hosting facilities) and a lower level of congestion at the expense of a lower net revenue receipt (when conversely a significant share of the revenues is utilized to decongestionate the hosting facilities).

The level of rigidities of external agents (namely, local residents and firms) in modifying the rate of allowed immigrants in the city (specific for each municipality) is captured by the constant parameter γ (>0)¹⁰, which reflects a marginal, increasing, and convex disutility cost in the rate at which the change in the number of allowed immigrants takes place. In this setting, the (stock) variable of the number of allowed immigrants might constitute a sluggish control variable since, as mentioned above, the variable over which the municipal authority has a better and direct control is represented by the the rate of change in the flow of allowed immigrants; therefore, following the approach suggested by Feichtinger et al. (1994), I assume that the rate of change in the flow of allowed immigrants would constitute a better control variable rather than its stock.

The squared term \(\dot {\psi (t)}\) implies that the disutility of both increases and decreases in the rate of change of allowed immigrants is weighted in the same way¹¹. Moreover, bigger variations in the rate of immigrant arrivals provoke a higher disutility with respect to smaller variations.

with r>0 representing the exogenous discount rate.

where the two costate variables, λ(t) and τ(t), represent, respectively, the shadow prices for the dynamics of the number of immigrants and the congestion level, denoting the marginal utility in relaxing the constraint (equivalently, the marginal disutility in tightening the constraint). Under the conditions guaranteeing the concavity of the Hamiltonian with respect to the control and state variables (see Appendix 1), the Mangasarian’s theorem assures the existence of an interior optimum.

with the limiting transversality conditions \({\lim }_{t \to +\infty } e^{rt} \tau (t) \pi (t) = 0\) and \({\lim }_{t \to +\infty } e^{rt} \lambda (t) \psi (t) = 0\) to assure a concave control problem. Equation (6) can also be rewritten as: \(\nu (t)=\frac {\lambda (t)}{\gamma }\). From this, it can be observed how ν(t) is decreasing in γ; indeed, the higher is the level of rigidities, the lower will be the the rate of change in the allowance of immigrants, and this is to compensate the disutility cost of rigidities.

Subsequently, setting all the equations from (7) to (10) equal to 0 describes the solution to the achievement of the steady state¹². First of all, from expression (7), it can be noticed that at the steady state, the rate of change of allowed immigrants in the city is null, that is ν ^∗=0 (which also implies λ ^∗=0). Secondly, from expression (8), it is possible to obtain ψ ^∗=ξ π ^∗; therefore, the level of allowance of immigrants at the steady state depends on the congestion level at the steady state together with the rate of expulsion of immigrants. This also means that at the steady state, higher levels in the number of immigrants are associated with higher congestion levels, and at the same time, a higher level of expulsions corresponds to a higher number of immigrants relocated to the hosting facilities (and vice versa for lower levels of ξ). Finally, the level of τ(t) at the steady state from Eq. (10) results to be: \(\tau ^{*}= - \frac {\Omega ^{\prime }(\pi ^{*} - \bar {\pi }^{*})}{\xi +r}\). Substituting then in expression (9), the values for λ ^∗=0 and τ ^∗, I derive: \(-U^{\prime }_{R}(\psi ^{*}) (\xi + r) + \Omega ^{\prime }(\pi ^{*} - \bar {\pi }^{*}) =0\), where UR′(ψ ^∗) and \(\Omega ^{\prime }(\pi ^{*} - \bar {\pi }^{*})\) are both positive; in order for the latter term to be positive to assure the equality, given that \(\frac {\partial \Omega }{\partial {(\pi ^{*})}} = \Omega ^{\prime }[\!\pi ^{*} - \bar {\pi }^{*}]\), it must be that \(\pi ^{*} - \bar {\pi }^{*}>0\); therefore, the optimal solution at the steady-state level corresponds to a situation of overcongestion. This leads to the following proposition:

From this, different equilibria in the path of convergence to the steady state can emerge, depending on the values of \(\left \|\mathcal {J}\right \|\) and k.

which is always positive.

is always negative.

In other words, if the level of rigidities expressed by the parameter γ is lower or equal to γ ₁ or higher or equal to γ ₂, there will be monotonic convergence, whereas if the value of γ lies between γ ₁ and γ ₂, there will be transient oscillations. This can be summarized in the following proposition:

Ultimately, the characterization of the dynamics towards the steady state will depend upon the level of rigidities, that hence help understanding the evolution of the current flows of immigrants who relocate to the Italian municipalities. For certain municipalities, where firms and local residents appear not to give particular importance to the immigration phenomenon, the level of rigidities towards variations in the rate of immigrant arrivals is reduced. As a consequence, these municipal authorities face a low level of pressure when setting or updating their desired immigration policy, so that the optimal number of immigrants who get relocated to the hosting facilities experiences a fast and monotonic increase in the process of convergence towards the steady state (for a graphical representation, see Appendix 2). Conversely, those municipalities facing higher levels of rigidities, in order to achieve the steady-state solution, are obliged to compromise continuously with firms and local residents, and the resulting bargaining process they have to go through provokes continuous alterations in the number of immigrants allowed to enter their jurisdiction. In other words, as the level of rigidities increases (i.e., both local residents and firms become more sensitive to variations in the rate of immigrant arrivals), the bargaining process that municipal authorities encounter gets stronger, and this lowers the margin of manoeuvre in adjusting their desired level of allowed immigrants, so that the rate of convergence slows down to compensate an increased disutility cost provoked by rigidities. Clearly, this leads to fluctuations in the process of convergence to the steady-state level of congestion as well as around it, until ultimately the steady-state level is reached. However, a different scenario could also emerge. Indeed, in the presence of a remarkably high level of rigidities, there will be a convergence towards the steady state following a dynamics similar to the former monotonic case. This happens because also the impact on the utility exercised by the levels of the number of immigrants and congestion increase with γ, and after a certain point, their level overcome the disutility cost of rigidities. That is to say, when the gap between optimal and current congestion levels of the immigration hosting facilities is too wide, the municipal authority needs to adjust rapidly the level of allowed immigrants in the city, in order to reduce the gap and eliminate a potential harmful risk due to an excessive level of under- or overcongestion; in this case, there will be a monotonic adjustment in the level of immigrants relocated to the hosting facilities. All these different scenarios can help explain the recent heterogeneity in immigration regulatory behavior emerging among different municipalities. It is also noteworthy to mention that the current heterogeneity of immigration policies among Italian municipalities did not lead to the emergence of a clear geographical cluster; indeed, within a same region (such as in Veneto or Lombardia, which in March 2016 were hosting, respectively, the 8 and 13% of the total number of immigrants in Italy), some municipalities were prompted to modify their acceptance rate to accommodate relevant number of immigrants without significant external pressure, whereas other municipalities were subject to high levels of rigidities by different parties, so that the acceptance rate of immigrants witnessed alternative phases (Linkiesta 2017).

¹ The vast majority of immigrants arriving in Italy holds a low level of education (Confindustria 2016).

² See Excelsior 2016.

³ For a detailed report inherent the demand for immigrant workers by Italian firms disaggregated at sector level and according to type of contract in the year 2015, see Excelsior 2015.

⁴ Following the adoption of the Pacchetto Immigrazione prepared by the Ministry of Interior in 2016, nowadays, immigrants hosted in municipal facilities waiting for the regularization of their residence permit (and, subsequently, for a hiring opportunity by firms) are employed by the municipality in socially useful works (in Italian, lavori socialmente utili (LSU)); the latter consist of activities of public interest regulated by the Italian law (e.g., street and park cleaning) and reserved to socially disadvantaged people.

⁵ The frequency with which a municipal authority decides to maintain or update the number of immigrants to relocate to its hosting facilities varies depending on municipality, but generally, it is about a few weeks. Furthermore, for security reasons, each time the municipality also needs the formal approval of the corresponding Prefettura, the authority representing the Ministry of Interior at the local level.

⁶ Every citizen has indeed the right to express his/her opinion on the immigration policy undertaken by the municipal authority.

⁷ Assume R as an undefined element of the utility function.

⁸ In this model, for the sake of simplicity, I assume homogeneity of inflowing immigrants. Nonetheless, this assumption might constitute a limitation, since immigrants could differ from each other in terms of mentality, prospects to assimilate, and above all, education level. To cope with this issue, one way could be, for instance, to differentiate between skilled and unskilled immigrants. Ultimately, however, the features of the immigration flows involving Italy could partially mitigate the potential issue of heterogeneity among immigrants. Indeed, as reported in the descriptive statistics above, the great majority of immigrants arriving in Italy after the immigration emergency is from Sub-Saharan African countries; they hence share a common cultural background and usually hold a poor or no education level. Additionally, also immigrants arriving from other countries such as Pakistan or Bangladesh in most of the cases always possess a low level of education (Confindustria 2016).

⁹ Another limitation of this model relies on the fact that important aspects such as the impact exercised by immigrants on the labor market and the role of both fiscal and market size effects of immigration are neglected. A more extensive analysis incorporating such issues could indeed provide a better and more sophisticated framework. Nonetheless, the contribution of immigrants to the total demand for low-skilled labor, although being important, remains reduced if compared to the overall low-skilled workforce employed by firms. Additionally, as expressed above, some bureaucratic barriers still lengthen the hiring process of immigrants by firms, in this way also contributing to increase the congestion level of municipal hosting facilities. As for the fiscal effects of immigration, despite the fact that some studies detected a positive impact of immigration on the welfare system of Western countries (see, e.g., Dustman and Frattini (2014)), other studies suggest that the same result could differ substantially depending upon the methodology employed, and thus the same final impact might be also considered minimal (Chojnicki 2013). The Italian case analyzed by this model depicts a reality in which the immigrants residing in the hosting facilities generally do not participate to the fiscal contribution (at least in the short run). The issue of market size effects of immigration raises a more complicated scenario, also due to the fact that the latter have been understudied in the literature; however, some of the most recent studies have highlighted how immigrants can positively affect the aggregate demand for goods (see, e.g., Aubry et al. 2016; Di Giovanni et al. 2014). In this model, for the sake of simplicity, the market size effects of immigration are neglected.

¹⁰ For the sake of simplicity, I assume that local residents and firms share the same level of rigidities; this means that in each municipality, their degree of sensitivity with respect to (opposite) variations in the rate of immigrant arrivals is the same.

¹¹ For the sake of simplicity, I assume that the (opposite) level of rigidities of firms and local residents are equally counterbalanced.

¹² The values at the steady-state level are indicated by the superscript asterisk (*).

¹³ The latter condition for oscillatory convergence, \(\left \|\mathcal {J}\right \|>\left (\frac {k}{2}\right)^{2}+r^{2}\frac {k}{2}\), is always verified, since k<0.