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 facilities^{8}. 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 jurisdiction^{9}.

*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.

Once they have been relocated to a municipality, immigrants are then subject to an additional check by the local security authorities and those who do not meet the requirements of asylum seekers are expelled. In light of these considerations, the relationship between the flow of immigrants entering and leaving the municipality and the dynamics of the congestion level of the hosting facilities at time *t*, *π*(*t*), can be synthesized by the following differential equation:

$$ \dot{\pi(t)} = \psi(t) - \xi \pi(t) $$

(1)

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.

Given all this, the maximization problem will consist of two state variables (the congestion level and the flow of immigrants relocated to the municipality) with their corresponding equation of motion and one control variable (the rate of change in the number of allowed immigrants). The utility of the municipal authority net of the the disutility costs of rigidities and congestion will be then:

$$ \hat{U_{t}} = U{_{R}}(\psi(t)) - \gamma \frac{\dot{\psi(t)}^{2}}{2} - \Omega [\!\pi(t) - \bar{\pi}(t)] $$

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 way^{11}. Moreover, bigger variations in the rate of immigrant arrivals provoke a higher disutility with respect to smaller variations.

The intertemporal utility function for the municipal authority considered in an infinite time horizon can subsequently be expressed as follows:

$$ \int_{0}^{\infty} e^{-rt} \left\{U{_{R}}(\psi(t)) - \gamma \frac{\dot{\psi(t)}^{2}}{2} - \Omega[\!\pi(t) - \bar{\pi}(t)]\right\}dt $$

(2)

with *r*>0 representing the exogenous discount rate.

Given a starting congestion level *π*(0)=*π*
_{0} and the dynamics of Eq. (1), a municipal authority has to choose the optimal path which maximizes its intertemporal utility function. In other words, the municipal authority’s problem becomes that one of choosing the optimal rate of change in the number of immigrants allowed in the city given the dynamics of the equations of motion. To complete the dynamic framework, let us label the control variable measuring the rate of change in the flow of allowed immigrants, \(\dot {\psi (t)}\), as *ν*(*t*), so that \(\dot {\psi (t)} = \nu (t)\). The intertemporal maximization problem can hence be written as:

$$ \operatorname*{max}_{\nu(t)} \int_{0}^{\infty} e^{-rt} \bigg\{U{_{R}}(\psi(t)) - \gamma \frac{{\nu(t)}^{2}}{2} - \Omega[\!\pi(t) - \bar{\pi}(t)]\bigg\}dt $$

(3)

$$ \begin{aligned} & \text{s.t.} & & \dot{\pi(t)} = \psi(t) - \xi \pi(t), \quad \psi(t)\geq 0, \quad \pi(t)>0 \\ & & & \dot{\psi(t)} = \nu(t) \\ & & & \pi(0) = \pi_{0} \\ & & & \psi(0) = \psi_{0} \\ \end{aligned} $$

(4)

From this, it is possible to derive the corresponding Hamiltonian function:

$$ \mathcal{H}(\psi, \pi, \nu, t) = U_{R}(\psi(t)) - \gamma \frac{\nu(t)^{2}}{2} - \Omega[\!\pi(t) - \bar{\pi}(t)] + \lambda(t) \nu(t) + \tau(t) (\psi(t) - \xi \pi(t)) $$

(5)

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.

From (5), the set of conditions for an optimal solution becomes:

$$\begin{array}{*{20}l} \partial_{\nu} \mathcal{H}(\psi, \pi, \nu, t) &= -\gamma \nu(t) + \lambda(t) \end{array} $$

(6)

$$\begin{array}{*{20}l} \dot{\psi(t)} &= \nu(t) \end{array} $$

(7)

$$\begin{array}{*{20}l} \dot{\pi(t)} &= \psi(t) - \xi \pi(t) \end{array} $$

(8)

$$\begin{array}{*{20}l} \dot{\lambda(t)} = r \lambda(t) - \partial_{\psi} \mathcal{H}(\psi, \pi, \nu, t) &= r \lambda(t) - [U^{\prime}_{R}(\cdot) +\tau(t)] \end{array} $$

(9)

$$\begin{array}{*{20}l} \dot{\tau(t)} = r \tau(t) - \partial_{\pi} \mathcal{H}(\psi, \pi, \nu, t) &= r \tau(t) - [-\Omega^{\prime}(\cdot) - \tau(t) \xi] \end{array} $$

(10)

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 state^{12}. 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:

###
**Proposition 1**

Under the conditions provided by expressions (3) and (4) for the resolution of the intertemporal optimization problem, the corresponding steady-state level always leads to a situation of overcongestion in the number of immigrants accommodated in the hosting facilities.

This result shows that the optimal solution corresponds to a situation of overcongestion in the number of immigrants hosted by the municipal authority, and this is independent from the level of rigidities and the share of the revenues which the municipality decides to utilize for increasing the optimal level of absorption of the hosting facilities. Thus, at the steady-state level, rigidities play no role, and the change in the rate of allowed immigrants in the city is null. However, the same level of rigidities does affect the process of convergence towards the steady state; indeed, rearranging the terms in the expressions (7)–(10) and substituting (6) into (7) and (10), it is possible to get:

$$\begin{array}{*{20}l} \dot{\psi(t)} &= \frac{\lambda(t)}{\gamma} \end{array} $$

(11)

$$\begin{array}{*{20}l}[-3pt] \dot{\pi(t)} &= \psi(t) - \xi \pi(t) \end{array} $$

(12)

$$\begin{array}{*{20}l}[-3pt] \dot{\lambda(t)} &= r \lambda(t) - U^{\prime}_{R}(\cdot) - \tau(t) \end{array} $$

(13)

$$\begin{array}{*{20}l}[-3pt] \dot{\tau(t)} &= \tau(t)(r+\xi) + \Omega^{\prime}(\cdot) \end{array} $$

(14)

Subsequently, consider the Jacobian matrix at the steady-state level constituted by expressions (11)–(14):

$$\begin{aligned} \mathcal{J}(\psi^{*}, \pi^{*}, \lambda^{*}, \tau^{*})& = \left[\begin{array}{llll} \frac{\partial\dot{\psi(t)}}{\partial\psi(t)} & \frac{\partial\dot{\psi(t)}}{\partial\pi(t)} & \frac{\partial\dot{\psi(t)}}{\partial\lambda(t)} & \frac{\partial\dot{\psi(t)}}{\partial\tau(t)} \\ \frac{\partial\dot{\pi(t)}}{\partial\psi(t)} & \frac{\partial\dot{\pi(t)}}{\partial\pi(t)} & \frac{\partial\dot{\pi(t)}}{\partial\lambda(t)} & \frac{\partial\dot{\pi(t)}}{\partial\tau(t)} \\ \frac{\partial\dot{\lambda(t)}}{\partial\psi(t)} & \frac{\partial\dot{\lambda(t)}}{\partial\pi(t)} & \frac{\partial\dot{\lambda(t)}}{\partial\lambda(t)} & \frac{\partial\dot{\lambda(t)}}{\partial\tau(t)} \\ \frac{\partial\dot{\tau(t)}}{\partial\psi(t)} & \frac{\partial\dot{\tau(t)}}{\partial\pi(t)} & \frac{\partial\dot{\tau(t)}}{\partial\lambda(t)} & \frac{\partial\dot{\tau(t)}}{\partial\tau(t)} \\ \end{array}\right]=\\ &=\left[\begin{array}{cccc} 0 & \quad 0 & \quad\frac{1}{\gamma} &\quad 0\\ 1 & \quad -\xi & \quad 0 &\quad 0\\ - U^{\prime\prime}_{R}(\cdot) & \quad 0 & \quad r & \quad -1\\ 0 & \quad \Omega^{\prime\prime}(\cdot) & \quad 0 & \quad r + \xi\\ \end{array}\right] \end{aligned} $$

The starting point to analyze the dynamics is to calculate the eigenvalues of the Jacobian at the steady state. Dockner (1985) proposes a formula to calculate the four eigenvalues (*E*
_{
i
},*i*=1,2,3,4) of any Jacobian associated to the canonical equations deriving from the first-order conditions of a two-dimensional optimal control model (i.e., two state and one control variable):

$$ E_{i} = \frac{r}{2} \pm \sqrt{\left(\frac{r}{2}\right)^{2} - \frac{k}{2} \pm \frac{1}{2} \sqrt{k^{2} - 4\parallel\mathcal{J}\parallel}} $$

(15)

where \(\left \|\mathcal {J}\right \|\) is the determinant of the Jacobian \(\mathcal {J}\) and *k* represents the sum of the determinants of the submatrices of \(\mathcal {J}\):

$$k:= \left\|\begin{array}{ll} \frac{\partial\dot{\psi}}{\partial\dot{\psi}} & \frac{\partial\dot{\psi}}{\partial\dot{\lambda}} \\ \frac{\partial\dot{\lambda}}{\partial\dot{\psi}} & \frac{\partial\dot{\lambda}}{\partial\dot{\lambda}} \end{array}\right\| + \left\|\begin{array}{ll} \frac{\partial\dot{\pi}}{\partial\dot{\pi}} & \frac{\partial\dot{\pi}}{\partial\dot{\tau}} \\ \frac{\partial\dot{\tau}}{\partial\dot{\pi}} & \frac{\partial\dot{\tau}}{\partial\dot{\tau}} \end{array}\right\| +2 \left\|\begin{array}{ll} \frac{\partial\dot{\psi}}{\partial\dot{\pi}} & \frac{\partial\dot{\psi}}{\partial\dot{\tau}} \\ \frac{\partial\dot{\lambda}}{\partial\dot{\pi}} & \frac{\partial\dot{\lambda}}{\partial\dot{\tau}} \end{array}\right\| $$

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*.

The determinant of the Jacobian is:

$$ \left\|\mathcal{J}\right\| = \frac{1}{\gamma}\bigg[\Omega^{\prime\prime}(\cdot) - U^{\prime\prime}_{R}(\cdot) \xi (\xi + r) \bigg] $$

(16)

which is always positive.

Then,

$$ k = \frac{U^{\prime\prime}_{R}(\cdot)}{\gamma} - \xi (\xi + r) $$

(17)

is always negative.

These results suggest that the dynamics of convergence can either follow a monotonic path of convergence \(\left (\text {if~} k<0 \text {~and~} 0<\left \|\mathcal {J}\right \|\leq \left (\frac {k}{2}\right)^{2}\right)\) or an oscillatory path \(\left (\text {if~} \left \|\mathcal {J}\right \|>\left (\frac {k}{2}\right)^{2} \text {and} \left \|J\right \|>\left (\frac {k}{2}\right)^{2}+r^{2}\frac {k}{2}\right)\)
^{13}. Ultimately, monotonic convergence implies \(\left (\frac {k}{2}\right)^{2} \geq \left \|\mathcal {J}\right \|\), whereas convergence through transient oscillations implies \(\left (\frac {k}{2}\right)^{2}<\left \|\mathcal {J}\right \|\) (Feichtinger et al. 1994). Deriving

$$ \bigg(\frac{k}{2}\bigg)^{2} - \left\|\mathcal{J}\right\| = \frac{\gamma^{2} (\xi+r)^{2} \xi^{2} - \gamma (4 \Omega^{\prime\prime}(\cdot) -2\xi(\xi+r)U^{\prime\prime}_{R}(\cdot))+U^{\prime\prime}_{R}(\cdot)^{2}} {4 \gamma^{2}}, $$

(18)

it is easy to notice that the denominator of expression (18) is always positive. The numerator provides a quadratic form with respect to *γ*. To study the sign of the numerator, consider when the latter is equal or greater than zero, which is when *γ*
^{2}(*ξ*+*r*)^{2}
*ξ*
^{2}−*γ*(4*Ω*
^{′′}(·)−2*ξ*(*ξ*+*r*)*UR*′′(·))+*UR*′′(·)^{2}≥0; the roots of the corresponding equation solved for *γ* are *γ*
_{1,2}=*Ξ*±*Θ*, with \(\Xi = \frac {2\Omega ^{\prime \prime }(\cdot)-\xi (\xi +r)U^{\prime \prime }_{R}(\cdot)}{\xi ^{2}(\xi +r)^{2}}\) and \(\Theta = 2\sqrt {\frac {\Omega ^{\prime \prime }(\cdot)^{2} - \xi (\xi +r) \Omega ^{\prime \prime }(\cdot) U^{\prime \prime }_{R}(\cdot)}{\xi ^{4}(\xi +r)^{4}}}\). Given that *Θ*>0 and *Ξ*>0, with |*Ξ*|>|*Θ*|, *γ*
_{1} and *γ*
_{2} are always distinct and positive. The set of solutions for the inequality is:

$$ \left\{\begin{aligned} &\left(\frac{k}{2}\right)^{2} - \left\|\mathcal{J}\right\| \geq 0 \quad \iff \quad \gamma \leq \Xi -\Theta \quad \text{or} \quad \gamma \geq \Xi + \Theta\\ &\left(\frac{k}{2}\right)^{2} - \left\|\mathcal{J}\right\|<0 \quad \iff \quad \Xi -\Theta < \gamma < \Xi + \Theta\\ \end{aligned}\right. $$

(19)

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:

###
**Proposition 2**

Under the conditions provided by expressions (3) and (4) for the resolution of the intertemporal optimization problem, there will be two different paths of convergence for the flow of immigrants and congestion level towards the steady state depending on the value assumed by *γ*. In particular, there will be monotonic convergence towards the steady state whenever *γ*∈[ 0,*γ*
_{1}]∪[ *γ*
_{2},*∞*). Conversely, if *γ*∈(*γ*
_{1},*γ*
_{2}) the convergence process towards the steady state will acquire an oscillatory behavior.

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).