Illegal immigration, deportation policy, and the optimal timing of return

Abstract

Countries with strict immigration policies often resort to deportation measures to reduce their stocks of illegal immigrants. Many of their undocumented foreign workers, however, are not deported but rather choose to return home voluntarily. This paper studies the optimizing behavior of undocumented immigrants who continuously face the risk of deportation, modeled by a stochastic process, and must decide how long to remain in the host country. It is found that the presence of uncertainty with respect to the length of stay abroad unambiguously reduces the desired migration duration and may trigger a voluntary return when a permanent stay would otherwise be optimal. Voluntary return is motivated by both economic and psychological factors. Calibration of the model to match the evidence on undocumented Thai migrants in Japan suggests that the psychological impact of being abroad as an illegal alien may be equivalent to as large as a 68 % cut in the consumption rate at the point of return.

Appendices

Appendix A: Optimal return with risk of deportation

1.1 A.1 Derivation of optimal consumption growth rate abroad

The Hamilton-Jacobi-Bellman equation is given by

$$ \rho V(a_{t})=\max\left\{u(c^{\ast}_{t})+\frac{\partial V(a_{t})}{\partial a_{t}}(r^{\ast}a_{t}+w^{\ast}-p^{\ast}c^{\ast}_{t})+\lambda\left( \tilde{V}(a_{t})-V(a_{t})\right) \right\}. $$

(15)

The first-order conditions with respect to $c_{t}^{\ast }$ and a _t yield

$$\begin{array}{@{}rcl@{}} u^{\prime}(c^{\ast}_{t})&-&p^{\ast}\frac{\partial{V_{t}}}{\partial{a_{t}}}=0, \end{array} $$

(16)

$$\begin{array}{@{}rcl@{}} \rho \frac{\partial{V_{t}}}{\partial{a_{t}}}&=&\frac{\partial^{2}{V_{t}}}{\partial{{a_{t}^{2}}}}\dot{a}_{t}+r^{\ast}\frac{\partial{V_{t}}}{\partial{a_{t}}}+\lambda\left( \frac{\partial{\tilde{V}_{t}}}{\partial{a_{t}}}-\frac{\partial{V_{t}}}{\partial{a_{t}}}\right). \end{array} $$

(17)

Differentiating Eq. 16 with respect to time and using the result in Eq. 17 yields

$$\frac{u^{\prime\prime}(c^{\ast}_{t})}{u^{\prime}(c^{\ast}_{t})}\dot{c}^{\ast}_{t}+r^{\ast}+\lambda\left( \frac{u^{\prime}(\tilde{c}_{t})p^{\ast}}{u^{\prime}(c^{\ast}_{t})p}-1\right)-\rho=0. $$

After rearranging terms and using $u^{\prime }(x)=(x)^{-\theta }$ ($x=\tilde {c}_{t},c^{\ast }_{t}$), we obtain

$$ \frac{\dot{c}^{\ast}_{t}}{c^{\ast}_{t}}=\frac{1}{\theta}\left\{\lambda\left[\left( \frac{\tilde{c}_{t}}{c^{\ast}_{t}}\right)^{-\theta}\frac{p^{\ast}}{p}-1\right]+r^{\ast}-\rho\right\}. $$

(18)

1.2 A.2 Derivation of optimal consumption growth rate after deportation

If U is deported at time ε, her objective is to maximize

$${\int}_{\varepsilon}^{T}u(\tilde{c}_{t})e^{-\rho(t-\varepsilon)}dt $$

subject to

$$\dot{a}_{t}=r a_{t}+w-p \tilde{c}_{t}, \ \ t\in[\varepsilon,T], \ \ a_{\varepsilon} \ \text{given,} \ a_{T}=0. $$

The solution to this problem is well known. The optimal growth rate of consumption is constant at $\frac {r-\rho }{\theta }$ and hence $\tilde {c}_{t}=\tilde {c}_{\varepsilon }e^{\frac {r-\rho }{\theta }(t-\varepsilon )}$. Using this in the differential equation for the asset position allows to solve for $\tilde {c}_{\varepsilon }$ (see Eq. 11 in the text) and subsequently for the time profile of assets:

$$a_{t}=a_{\varepsilon}e^{r(t-\varepsilon)}+ w\frac{e^{r(t-\varepsilon)}-1}{r}-p\tilde{c}_{\varepsilon}\frac{e^{\frac{r-\rho}{\theta}(t-\varepsilon)}-e^{r(t-\varepsilon)}}{g}, \ \ t\in[\varepsilon,T]. $$

Appendix B: Alternative functional forms

1.1 B.1 Other forms of homesickness function

The results presented in Section 4.3 are undoubtedly sensitive to some extent to the chosen specification of the homesickness function. Suppose, for example, that homesickness increases with time spent abroad following a linear trend or increases at a decreasing rate or even declines over time due to better assimilation. The last case is, however, highly unlikely for illegal Thai migrants in Japan as they typically do not learn the Japanese language and restrain from being exposed to the public.

We consider here two alternative specifications of the h _t function in Eq. 14: linear (h _t = α t) and concave (h _t = t ^α−1), where the latter captures decaying marginal homesickness effect. In the absence of homesickness (α = 0), both functions are normalized to zero. For each functional form, the parameter α is extracted from the data on undocumented Thai migrants in Japan and the results are summarized in Tables 3 and 4. The predicted consumption-equivalent loss is reported in columns 3 and 4 for alternative values of θ and two consumption reference points: the 30th month (half average trip duration) and the 60th month.

Table 3 Concave homesickness function, h _t = t ^α−1

Table 4 Linear homesickness function, h _t = α t

If we consider the 30th month as the point of reference, then the effect of homesickness is equivalent to a 33.65 % (for θ = 0.95) to 55.6 % (for θ = 0.75) cut in the migrant’s consumption rate in that month when h _t is exponential, a 38.84 to 65.6 % cut when h _t is linear, and a 53.94 to 83.1 % cut when h _t is concave. Logically, the convex functional form yields a relatively smaller loss than the linear form which in turn yields a smaller loss than the concave function. If we consider the last month of the trip as the point of reference, then the effect of homesickness translates into approximately a 62 % (θ = 0.95) to 89 % (θ = 0.75) decline in the consumption rate for all the three specifications. Thus, regardless of the chosen functional form, homesickness remains a powerful element influencing return decisions of undocumented foreign workers, as suggested by descriptive studies (see, e.g., Kibria 2004).

We can also compare the effectiveness of a stricter deportation policy in reducing migration duration under alternative specifications of the h _t function. Columns 6 and 7 of Tables 3 and 4 report the reduction in τ following a twofold and, respectively, a threefold increase in the deportation rate. The concave h _t function captures a relatively fast-growing negative impact of homesickness at the beginning of the trip and a subsequent slow-down. The linear function captures a steady increase, while the convex function captures a slowly growing homesickness at the beginning of the trip but an explosively high growth at longer durations. The effect of an increase in the deportation rate on the optimal-return date is thus more pronounced for a concave specification (regardless of θ) since the marginal impact of homesickness (the slope of h _t ) is larger for t<τ. For instance, a doubling of λ reduces τ by 2.81 months if h _t is concave and by only 1.82 months if it is convex (assuming θ = 0.95).

1.2 B.2 A more general utility function

So far, we have presented a model where homesickness entered a migrant’s utility function in an additively-separable fashion. It was useful to adopt separability because in this case, the optimal consumption growth rate was not affected by the time spent in the host country. As noted by an anonymous referee, one should consider alternative ways of combining consumption and homesickness in the utility function. In this appendix, we propose a more general specification to allow for the marginal utility of consumption to depend on homesickness, denoted by H _t . This implies that the value function in the HJB equation depends not only on the asset position but also on time. Consequently, the expression for the optimal consumption growth rate has extra terms. Formally, the HJB equation now reads:

$$\begin{array}{@{}rcl@{}} \rho V(a_{t},t)\!&=&\!\max\left\{u(c^{\ast}_{t},H_{t})+\frac{\partial V(a_{t},t)}{\partial a_{t}}(r^{\ast}a_{t}+w^{\ast}-p^{\ast}c^{\ast}_{t})+\lambda\left( \tilde{V}(a_{t})-V(a_{t})\right)\right.\\&&\left.\quad\qquad+\frac{\partial V(a_{t},t)}{\partial t} \right\}. \end{array} $$

The first-order conditions are given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial u(c^{\ast}_{t},t)}{\partial c^{\ast}_{t}}&-&p^{\ast}\frac{\partial{V}}{\partial{a_{t}}}=0, \\ \rho\frac{\partial{V}}{\partial{a_{t}}}&=&\frac{\partial^{2}{V}}{\partial{{a_{t}^{2}}}}\dot{a}_{t}+r^{\ast}\frac{\partial{V}}{\partial{a_{t}}}+\lambda\left( \frac{\partial{\tilde{V}}}{\partial{a_{t}}}-\frac{\partial{V}}{\partial{a_{t}}}\right)+\frac{\partial^{2} V}{\partial t\partial a_{t}}. \end{array} $$

Differentiation of the first condition yields:

$$u_{cc}^{\ast}\dot{c}^{\ast}+u^{\ast}_{ct}=p^{\ast}[V_{aa}\dot{a}+V_{at}], $$

where we used short-hand notation for the partial derivatives $u_{cc}^{\ast }\equiv \frac {\partial ^{2} u(c^{\ast }_{t},t)}{\partial c^{*2}_{t}}$ and similarly $u_{ct}^{\ast }\equiv \frac {\partial ^{2} u(c^{\ast }_{t},t)}{\partial c^{\ast }_{t}\partial t}$. Combining this with the second condition yields:

$$u_{cc}^{\ast}\dot{c}^{\ast}+u^{\ast}_{ct}=-u^{\ast}_{c}\left\{r^{\ast}-\rho+\lambda\left[\left( \frac{\tilde{u}_{c}}{u^{\ast}_{c}}\right)\frac{p^{\ast}}{p}-1\right] \right\} $$

$$\frac{\dot{c}^{\ast}}{c^{\ast}}=-\frac{u^{\ast}_{c}}{c^{\ast}u_{cc}^{\ast}}\left\{r^{\ast}-\rho+\lambda\left[\left( \frac{\tilde{u}_{c}}{u^{\ast}_{c}}\right)\frac{p^{\ast}}{p}-1\right] \right\}-\frac{u^{\ast}_{ct}}{c^{\ast}u_{cc}^{\ast}} $$

We can immediately see that time spent abroad affects the growth rate of consumption through two channels. One is represented by the last negative term (recall that $u^{\ast }_{ct}<0$) and the other is the indirect effect through the marginal utility of consumption in the term multiplying λ. Both effects work to speed up consumption growth: the longer the migrant stays abroad, the higher is the growth rate of consumption.

Let us consider a specific example where consumption and homesickness enter multiplicatively in the utility function. For instance, let $u(c^{\ast },H_{t})=\frac {c^{*1-\theta }_{t}}{1-\theta }H_{t}^{-1}$, where H _t =1 + h _t . Note that we added 1 to the function h because now it enters utility multiplicatively, and therefore, we need to ensure that when the homesickness parameter α is set to zero, the utility of consumption is not affected by the homesickness component. Earlier, we considered three types of h _t function (exponential, linear, and concave), which were normalized to zero when α = 0. In the current setup, they need to be normalized to unity and this is why we now work with the function H _t and not h _t .

The above functional form of u(c ^∗,H _t ) has the desired properties, in the sense that it is increasing and concave in consumption (standard CRRA), decreasing in homesickness (for θ∈[0,1]) and the marginal utility of consumption is decreasing in homesickness as well. The three specifications of the H _t function—exponential (H _t = e ^αt), concave (H _t = t ^α), and linear (H _t =1 + α t)—result in three different expressions for the optimal consumption growth rate abroad:

$$\begin{array}{@{}rcl@{}} && g^{\exp}=\frac{1}{\theta}\left\{r^{\ast}-(\rho+\alpha)+\lambda \left[\frac{p^{\ast}}{p}\left( \frac{\tilde{c}}{c^{\ast}}\right)^{-\theta}e^{\alpha t}-1\right]\right\},\\ && g^{\text{con}}=\frac{1}{\theta}\left\{r^{\ast}-\rho+\lambda\left[\frac{p^{\ast}}{p}\left( \frac{\tilde{c}}{c^{\ast}}\right)^{-\theta}t^{\alpha} -1\right] -\frac{\alpha}{t}\right\},\\ &&g^{\text{lin}}=\frac{1}{\theta}\left\{r^{\ast}-\rho+\lambda\left[\frac{p^{\ast}}{p}\left( \frac{\tilde{c}}{c^{\ast}}\right)^{-\theta} (1+\alpha t)-1\right] -\frac{\alpha}{1+\alpha t}\right\}. \end{array} $$

Note that with the exponential specification, the intensity of homesickness essentially alters the subjective rate of time preference from ρ to ρ + α.

Let us now turn to the numerical results. We are primarily interested in exploring robustness of our findings presented in Section 4 with respect to alternative specifications of the utility function. Given that time spent abroad increases the optimal consumption growth rate abroad, we expect that the consumption-equivalent loss due to homesickness is larger with a multiplicative utility function than with an additive function. Tables 5 to 7 below replicate Tables 2, 3, and 4 assuming that $u(c^{\ast }_{t},H_{t})=\frac {c_{t}^{*1-\theta }}{1-\theta }H_{t}^{-1}$, where the homesickness function H _t takes three alternative forms, specified in the caption of each table.³²

Table 5 Exponential homesickness function, H _t = e ^αt

Table 6 Concave homesickness function, H _t = t ^α

Table 7 Linear homesickness function, H _t =1 + α t

The quantitative conclusions which arise from the model with a multiplicative utility are very similar to those arising from the model with an additive utility: (1) Considering the fifth column of Tables 5, 6, and 7, we see that when the homesickness function is concave, tightening of the deportation policy (increase in λ) is more effective than when H is linear, which in turn is more effective than when H is exponential. (2) The larger is the elasticity of intertemporal consumption substitution, the smaller is the policy impact on migration duration. (3) We see in the fourth column of Tables 5, 6, and 7 that consumption-equivalent loss due to the psychological effect, evaluated at the 60th month abroad, is quantitatively similar across the three functional forms capturing homesickness. These are essentially the same conclusions which we have drawn from our baseline model with additive utility.

Comparing the results stemming from an additive utility model to those stemming from a multiplicative-utility model , we find no major differences. The quantitative impact of a change in λ is very similar. For instance, when homesickness is represented by an exponential function, a doubling of λ results in a 2.07 % decline in τ with additive utility and in a 2.87 % decline with a multiplicative utility, for θ = 0.75; a 3.01 % and a 3.40 % decline, respectively, for θ = 0.9; and in a 3.03 and 3.66 % decline, respectively, for θ = 0.95. When homesickness is represented by a linear function, the numbers are 2.59 and 2.94 % for θ = 0.75, 3.40 and 3.46 % for θ = 0.9, and 3.66 and 3.69 % for θ = 0.95. The numbers are also very similar in the case of a concave function. Not surprisingly, with the multiplicative structure τ declines slightly more than with additive preferences.

There is one dimension, however, along which the additive and the multiplicative-utility models produce slightly different results. This occurs when we calculate the fall in the consumption rate which would replicate the homesickness effect on welfare (shown in columns 3 and 4). For instance, the additive utility model with exponential h produces a 89.8 % drop in c ^∗ for θ = 0.75, while in the multiplicative case, we obtain a 93.76 % drop. As θ increases, the percentage difference becomes less pronounced: a 68.32 % drop vs. 69.20 % for θ = 0.9 and a 62.53 vs. 62.90 % for θ = 0.95. With linear h, we have 89.86 vs. 93.40 % for θ = 0.75, 68.37 vs. 69.20 % for θ = 0.9, and 62.57 vs. 62.90 % for θ = 0.95. The multiplicative specification produces larger percentage fall in consumption compared to the additive function because in the former case homesickness has a negative impact not only on the welfare level but also on the marginal utility of consumption. This is especially relevant when the elasticity of marginal utility (θ) is small. This result is quite intuitive. For an identical drop in consumption, the marginal utility increases by less in the multiplicative case. Thus, the decline in consumption should be larger in order to counteract the negative effect of homesickness, and this is especially so when the marginal utility is less elastic (small θ).

Overall, we can safely conclude that changing the preference structure from additive to multiplicative does not significantly alter our qualitative or even quantitative results. On the qualitative side, we note that with the latter preferences the optimal migration duration is shorter and the effect of a tighter deportation policy on the length of the trip is stronger. On the quantitative side, we find that the differences are fairly marginal, especially if we focus on a relatively elastic utility of consumption.

Appendix C: Homesickness and migrants’ interactions

As pointed out by an anonymous referee, decisions of one individual migrant may influence the decisions of another. Along those lines, one can think of extending the present model to capture the notion that homesickness of a single individual may not only be a function of the amount of time spent abroad but also a function of the number of migrants from the same source country residing in the host country. For the sake of argument, let us focus on a two-country model with one host- (Japan) and one source-country (Thailand) and let S _t denote the stock of undocumented Thai migrants in Japan at time t. Suppose that homesickness is a decreasing function of this stock, i.e., ∂ h(t,S _t )/∂ S _t <0. Let us assume, for example, that

$$h(t,S_{t})=(e^{\alpha t}-1)S_{t}^{-\gamma}, $$

where the parameter −γ is the elasticity of h _t with respect to S _t . In general, one can think of γ as being migrant-specific.

Given the deportation rate λ, the number λ S _t of undocumented Thai workers are deported on average every period. At the same time there is an inflow of new arrivals and an outflow of those who are voluntarily returning. Let us denote the difference between the arrivals and voluntary departures (as a proportion of the stock) by the net arrival rate β(λ). As an increase in the deportation rate has a deterrent effect on new inflows and a stimulating effect on voluntary outflows, we can assume that $\beta ^{\prime }(\lambda )<0$. Thus, the change of the stock of migrants over time can be described by the following differential equation:

$$\dot{S}_{t}=\left[\beta(\lambda)-\lambda\right]S_{t}, $$

so that an increase or a decrease of the stock occurs depending on whether β(λ) is greater or smaller than λ. The time-profile of the stock is then $S_{t}=S_{0}e^{\left [\beta (\lambda )-\lambda \right ]t}$, where S ₀ stands for the initial migrants’ stock. We can then write the homesickness function as

$$h_{t}=(e^{\alpha t}-1)S_{0}^{-\gamma}e^{-\gamma\left[\beta(\lambda)-\lambda\right]t}. $$

It follows that

$$\frac{\partial h_{t}}{\partial \lambda}=(e^{\alpha t}-1)S_{0}^{-\gamma}e^{-\gamma\left[\beta(\lambda)-\lambda\right]t}\gamma\left[1-\beta^{\prime}(\lambda)\right] t>0, \ \ \forall t>0. $$

A more vigorous deportation policy now affects a single migrant’s behavior through two channels. First, through the channel which I emphasize in the paper and second, through an additional channel which arises because the change in the deportation policy has a negative effect on the total stock of migrants. This contributes to an increase in homesickness (greater h), which in turn induces an undocumented migrant to advance even further the date of voluntary return.

Some interesting policy implications follow immediately from this discussion: (a) If a host country allows the stock of undocumented workers to accumulate, it makes any deportation policy, described by the rate λ, less effective in inducing voluntary return. Migrants are simply less homesick when there is a larger stock of their compatriots in the host country facing the same predicament. (b) In a setting with multiple source countries, any given deportation rate is more effective in inducing voluntary return of illegal aliens from countries with a smaller stock of nationals in the host country. A smaller stock makes them more homesick and hence more inclined to voluntarily return to their country of origin relatively sooner.