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Legal and illegal immigrants: an analysis of optimal saving behavior

Abstract

Savings of guest workers as well as of undocumented migrants represent important inflows of foreign exchange for some developing countries. This paper compares the saving behavior of these two types of migrants, assuming that the former are authorized to work abroad for a specific period of time, while the latter can stay until apprehended and deported by the immigration authorities. Due to the risk of deportation, the saving rate of an illegal immigrant is found to be initially above that of a documented migrant. This precautionary saving phenomenon is, however, short-lived. A key finding of the paper is that the total repatriated assets of an illegal migrant are always lower than those of a documented worker, provided that their duration of stay abroad is identical. This is because the undocumented migrant’s saving rate falls over time as her expected lifetime earnings are adjusted upwards every day that she avoids apprehension.

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Notes

  1. Each year, the stock of undocumented migrants in the EU is estimated to be growing by 500,000 individuals (IOM 2003). Inflows of similar magnitude are reported for the USA, where the stock of undocumented immigrants was estimated at roughly 10.8 million in the first quarter of 2009 (Center for Immigration Studies 2009).

  2. In 2008, 192 million foreign workers, including those without proper documentation, sent $328 billion from the developed to the developing countries. This represents almost three times the amount of official aid flows from OECD member states (World Bank 2009).

  3. See World Bank, http://blogs.worldbank.org/peoplemove/remittance-flows-to-developing-countries.

  4. See Adams (1991), Durand et al. (1996), Lucas (2005), Massey and Parrado (1998), McCormick and Wahba (2001), and Taylor (1987).

  5. This is the structure of typical guest-worker programs operated in Taiwan, South Korea, and Singapore, with durations of stay limited to 2–5 years. Contract-completion clauses in guest-worker contracts of numerous host economies in Asia allow (in some cases require) employers to withhold a part of a worker’s salary until the time of departure. This serves to prevent contract workers from remaining in the host country illegally. The seasonal guest-worker programs in Western Europe and North America typically allow for permits valid for less than a year.

  6. See, e.g., Skinner (1988), Toche (2005), Wälde (1999), and Zeldes (1989), to mention just a few.

  7. See, e.g., Djajić (1989).

  8. See Djajic (2010), Djajic and Milbourne (1988), Dustmann (1995, 1997), Kirdar (2010), and Mesnard (2004).

  9. The probability of deportation is also present in the models of Woodland and Yoshida (2006), Djajić (2011) and Auriol and Mesnard (2012). In these papers, however, deportation is a zero-one event happening at the border of the host country. If the migrant is lucky not to get caught at the border, she can work freely in the host country and no longer face any uncertainty. In the present model, the migrant continuously faces the risk of deportation while working in the host country, a feature which fundamentally affects her optimal consumption-saving behavior. Djajić (2013) and Djajić and Vinogradova (2013) consider the possibility of a migrant being subject to deportation while working in the underground economy. In contrast with the present study, however, they treat the problem deterministically.

  10. I assume that initial assets are large enough to cover migration costs, i.e., \(a_{0}\geq 0\). I therefore rule out the case of borrowing to finance migration. On this issue, see Djajić and Vinogradova (2013).

  11. The derivations of all the equations are relegated to the Appendix 1.

  12. To make the comparison as clear as possible, I assume that all savings are repatriated at the point of return, regardless of whether the migrant is a contract worker or undocumented alien. Remittances are not modeled explicitly in this paper.

  13. Toche (2005) obtains a similar result in the context of a model with a random employment status; see also Wälde (1999).

  14. See Jones and Pardthaisong (1999), Sobieszczyk (2000). Amuedo-Dorantes et al. (2005) report that the average age of Mexican migrants to the USA in the Mexican Migration Project (MMP93) was 33 years, and their average length of stay was close to 3 years.

  15. To my knowledge, there are no empirical studies which attempt to estimate \(\theta \) for illegal (or even legal) migrants. Vissing-Jørgensen (2002) estimates EICS for stock- and bondholders, distinguishing among three wealth groups, as well as for non-stockholders. Her estimates range from 0.29 for stockholders to 1.38 for bondholders with higher estimates for top wealth layer households and close to zero estimates for non-stockholders. See also Epstein and Zin (1991), Hansen and Singleton (1982), and Keane and Wolpin (2001). I rely on these estimates for calibrating \(\theta \) in my quantitative analysis.

  16. With \(\delta =r\) the time path of a guest worker’s consumption is flat at the level \(c_{0}\), while that of the undocumented migrant is upward sloping when she is located abroad and flat after deportation. When \(\delta <r\) (\(\delta >r\)), the time paths of consumption of both G and U rotate counterclockwise (clockwise), which results in a larger (smaller) amount of savings repatriated to the source country.

  17. Note that the longer U manages to remain abroad, the shorter is the second, source-country phase of her planning horizon. As the duration of the first phase expands and that of the second phase contracts, with U successfully avoiding apprehension, it eventually becomes optimal to repatriate a smaller stock of savings by actually dissaving while abroad.

  18. Since G does not face any uncertainty with respect to her duration of stay abroad, her actual and expected RA are identical. Due to the risk of deportation, however, those of U are not.

  19. Given that the event of deportation follows the Poisson process, the waiting time until deportation is an exponentially distributed random variable. The truncation is necessary since the migrant’s planning horizon is finite (equal to T). With an infinite horizon, the density is just the numerator of \(f_{s}\).

  20. For very long expected durations of stay abroad (\(\lambda \rightarrow 1/T\)) and the corresponding lengths of the permit (\(\tau \rightarrow T\)), G behaves as a permanent migrant with her repatriated savings approaching zero, while U’s expected RA are positive due to the presence of deportation risk. These values of \(\tau \) are not empirically relevant, and so they do not merit further discussion in the paper.

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Correspondence to Alexandra Vinogradova.

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Appendices

Appendix 1: Solution to a guest worker’s and an illegal worker’s optimization programs

Legal guest worker

The objective is to maximize

$$V^{G}=\int_{0}^{T}u(c_{t})e^{-\delta t}\mathrm{d}t, $$

subject to the budget constraint

$$ \int_{0}^{\tau}(w^{*}-c_{t})e^{-r t}\mathrm{d}t+a_{0}+\int_{\tau}^{T}(w-c_{t})e^{-rt}\mathrm{d}t=0. $$
(10)

The Lagrangian function is given by

$$L=\int_{0}^{T}u(c_{t})e^{-\delta t}\mathrm{d}t +\mu\left[ \int_{0}^{ \tau}(w^{*}-c_{t})e^{-rt}\mathrm{d}t+a_{0}+\int_{\tau}^{T}(w-c_{t})e^{-rt}\mathrm{d}t\right] $$

and the first-order condition with respect to consumption choice

$$ \frac{\partial L}{\partial c_{t}}=u^{\prime}(c_{t})e^{-\delta t}-\mu e^{-rt}=0. $$
(11)

Equation (11) implies that consumption rate is equal to \(c_{t}=c_{0}e^{ \frac {r-\delta }{\theta }t}\), with \(c_{0}=\mu ^{-1/\theta }\) and where we used the iso-elastic utility specification \(u(x)=\frac {x^{1-\theta }}{1-\theta }\). Using this in the budget constraint (10), we obtain

$$\int_{0}^{\tau}\left(w^{*}-c_{0}e^{\frac{r-\delta}{\theta}t}\right)e^{-rt}\mathrm{d}t+a_{0}+ \int_{\tau}^{T}\left(w-c_{0}e^{\frac{r-\delta}{\theta}t} \right)e^{-rt}\mathrm{d}t=0. $$

Solving for \(c_{0}\), we obtain Eq. (2) in the text.

Illegal immigrant

The problem of a migrant facing a risk of deportation is a stochastic optimal control problem which can be addressed by writing the Hamilton–Jacobi–Bellman equation:

$$ \max \quad \left\{u\left(c^{u}_{t}\right)+\frac{\partial V_{t}}{\partial a_{t}} \left(ra_{t}+w^{*}-c^{u}_{t}\right)\right\}+\lambda\left(V^{d}_{t}-V_{t}\right)- \delta V_{t}=0, $$
(12)

where the superscript d stands for “deportation,” and \(V_{t}\) is U’s value function. The first-order conditions with respect to \(c_{t}^{u}\) and \(a_{t}\) yield

$$ u^{\prime }\left(c^{u}_{t}\right)-\frac{\partial{V_{t}}}{\partial{a_{t}}}=0, $$
(13)
$$ \frac{\partial^{2}{V_{t}}}{\partial{a_{t}^{2}}}\dot{a}_{t}+r\frac{ \partial{V_{t}}}{\partial{a_{t}}}+\lambda\left( \frac{\partial{V_{t}^{d}}}{ \partial{a_{t}}}-\frac{\partial{V_{t}}}{\partial{a_{t}}}\right)-\delta \frac{ \partial{V_{t}}}{\partial{a_{t}}}=0. $$
(14)

Differentiating (13) with respect to time and using the result in Eq. (14) yields

$$ \frac{u^{\prime \prime }(c^{u}_{t})}{u^{\prime }(c^{u}_{t})}\dot{c}^{u}_{t}+r+\lambda \left(\frac{u^{\prime}\left(c_{t}^{d}\right)}{u^{\prime }(c^{u}_{t})}-1\right)-\delta=0. $$

After rearranging terms and using \(u^{\prime }(c^{i}_{t})=(c^{i}_{t})^{-\theta}\) ( \(i=d,u\)) we obtain

$$ \frac{\dot{c}^{u}_{t}}{c^{u}_{t}}=\frac{1}{\theta}\left\{\lambda\left[\left( \frac{c_{t}^{d}}{c^{u}_{t}}\right)^{-\theta}-1\right]+r-\delta\right\}. $$
(15)

Note that the term in the square brackets is unambiguously positive as the consumption rate in deportation, \(c_{t}^{d}\), is always smaller than \( c_{t}^{u}\), otherwise migration would not have taken place. Thus, the ratio \( c_{t}^{d}/c^{u}_{t}\) raised to a negative power is always greater than unity.

It is obvious from the above equation that the solution depends on the migrant’s consumption in “deportation,” \(c^{d}_{t}\). But \(c^{d}_{t}\) can be easily obtained by solving the deterministic optimization problem of an individual who is deported at an arbitrary time, say \(\xi \in [0,T]\). His objective is to maximize

$$\int_{\xi}^{T}u\left(c_{t}^{d}\right)e^{-\delta (t-\xi)}\mathrm{d}t $$

subject to

$$ \dot{a}_{t}=r a_{t}+w-c_{t}^{d}, $$
(16)

the terminal condition \(a_{T}=0\) and the initial condition given by \(a_{\xi }\), i.e., the amount of assets accumulated abroad up to time \(\xi \) which the migrant brings with him to the source country at the time of deportation.

The present value Hamiltonian is

$$H=u\left(c_{t}^{d}\right)e^{-\delta (t-\xi)}+\nu_{t}\left[r a_{t}+w-c_{t}^{d}\right], $$

where \(\nu _{t}\) is the co-state variable, and the first-order conditions are

$$ \frac{\partial H}{\partial c_{t}^{d}}=0 => \quad u^{\prime }\left(c_{t}^{d}\right)e^{-\delta (t-\xi)}=\nu_{t} $$
(17)
$$ \frac{\partial H}{\partial a_{t}}=-\dot{\nu}_{t} => \quad r\nu_{t}=-\dot{\nu}_{t} $$
(18)

Taking the time derivative of Eq. (17) and using the result in Eq. (18), we obtain the usual Ramsey type condition for consumption growth rate

$$\frac{\dot{c}^{d}_{t}}{c_{t}^{d}}=\frac{r-\delta}{\theta}, \quad t\in[ \xi_{+},T]. $$

This equation implies the following consumption path

$$c_{t}^{d}=c^{d}_{\xi}e^{\frac{r-\delta}{\theta}(t-\xi)}, $$

where \(c^{d}_{\xi }\) is determined by solving the differential equation for asset accumulation (16):

$$ c^{d}_{\xi}=\left[a_{\xi}+\frac{w}{r}\left(1-e^{-r (T-\xi)}\right)\right] \frac{g}{e^{g(T-\xi)}-1}, $$
(19)

Now, Eq. (19) can be substituted in Eq. (15) to yield the law of motion for the illegal immigrant’s consumption. The next step is to solve the system of two differential equations, one for consumption and the other for assets accumulation, which is done numerically.

Appendix 2: Robustness checks

This Appendix examines the sensitivity of the results to various model specifications.

Unequal wages for legal and illegal workers

I denote the foreign legal wage of the guest worker by \(w^{*G}\) and the underground wage of the undocumented worker by \(w^{*U}\) and show their combinations such that repatriated assets of the two types of migrants are identical. Figure 7 shows two upward sloping schedules, one for the benchmark case \(\tau =4\) and \(\lambda =0.25\) (solid line) and the other for \(\tau =10\) and \(\lambda =0.1\) (dashed line). Note that the slopes of both schedules are, as expected, less than unity. That is, for the expected repatriated assets to be identical for the two types of migrants, the wage of the legal worker should be lower than the underground wage. The larger the host-country underground wage, the wider the discrepancy should be.

Fig. 7
figure 7

Combinations of \(w^{*G}\) and \(w^{*U}\) such that \({\rm ERA}^{{\rm U}}={\rm RA}^{{\rm G}}\)

For example, in the benchmark case (see solid line), if \(w^{*U}=2\), then \(w^{*G}\) must be around 1.8, or 10 % lower, but if \(w^{*U}=10\), then \(w^{*G}\) must be around 8.64, or 13.6 % lower, in order to keep \({\rm RA}^{{\rm G}}={\rm ERA}^{{\rm U}}\). Anywhere above (below) the \({\rm RA}^{{\rm G}}={\rm ERA}^{{\rm U}}\) schedule, a guest worker repatriates more (less) than an undocumented worker. In addition, the longer the expected duration of stay abroad, the smaller should the legal wage be in order to keep \({\rm RA}^{{\rm G}}={\rm ERA}^{{\rm U}}\). This is illustrated by the dashed schedule (longer \(\tau \)) which lies below the solid schedule (short \(\tau \)). The intuition is straightforward: When the expected duration of stay abroad is relatively long, the second phase of the planning horizon, i.e., back home, becomes relatively short. An illegal migrant will accumulate smaller savings than in the case of a short expected duration. Therefore, the legal wage must be even smaller for the repatriated assets to remain equalized.

Planning horizon

The effect of extending migrants’ planning horizon from \(T=30\) years to \(T=50\) years, while keeping the other parameters at their benchmark values and \(\lambda =0.1\), is shown in Fig. 8.

Fig. 8
figure 8

Longer planning horizon, T

A longer planning horizon obviously calls for an increase in the saving rate while abroad for both types of migrants (see thin lines) since the savings now need to finance home-country consumption over a longer period of time. The precautionary saving motive during an initial phase of foreign stay in the case of the illegal migrant is still present. As in the benchmark, her saving rate falls over time and by \(t=1/\lambda =10\) is way below the legal guest worker’s rate.

Figures 9 and 10 show what happens to the time paths of the saving rate and the asset position when the interest earned on savings differs from the migrants’ rate of time preference (benchmark is shown by bold curves). When \(r>\delta \), the consumption rate of G follows the Keynes-Ramsey rule growing at \(\frac {r-\delta }{\theta }>0\), and thus the saving rate declines over time, as illustrated in Fig. 9a by the thin solid line. The saving rate of U, shown by the thin dashed line, falls at even higher rate as compared to the benchmark case \(r=\delta \) because consumption growth rate is faster (recall Eq. (9)).

Fig. 9
figure 9

Optimal paths of savings and asset position, \(r=6\) %, \(\delta =3\) %

When \(r<\delta \), the opposite occurs: consumption rates of both U and G decline over time, while the saving rates increase. Relaxing the assumption \(r=\delta \) has therefore no bearing on the key results. The precautionary motive remains valid, so as the fact that the saving rate of U ends up being below that of G at the end of their foreign stay. Figures 9b and 10b clearly demonstrate that the total repatriated assets of U (red line) are smaller than those of G (black line) at the time of return.

Fig. 10
figure 10

Optimal paths of savings and asset position, \(r=0.5\) %, \(\delta =3\) %

Figure 11 illustrates the effect of an increase in the wage differential from \(w^{*}/w=2\) (in the benchmark case, bold lines) to \(w^{*}/w=4\) (thin lines) on the saving rates and assets of U and G. Clearly, with a higher wage abroad, the saving rates while in the host country increase, and the dissaving after return is also higher since both migrants have accumulated a larger stock of assets as compared to the benchmark case.

Fig. 11
figure 11

Larger wage differential, \(w^{*}/w\)

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Vinogradova, A. Legal and illegal immigrants: an analysis of optimal saving behavior. J Popul Econ 27, 201–224 (2014). https://doi.org/10.1007/s00148-013-0481-9

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Keywords

  • Illegal immigration
  • Uncertainty
  • Saving behavior

JEL Classifications

  • D81
  • F22
  • J61