Remittances, savings and return migration under uncertainty

Lemma 1.

In period 2, the migrant returns to the origin country if D _w <D _R .

Having determined the optimal location condition, we can now start the analysis of the migrant’s problem in period 1. In this period, wages, and consequently the wage gap D _w , are treated as random variables. Return migration is therefore uncertain, and depends on the distribution of D _w . Let f(D _w ) denote the marginal density of D _w , where $f\left(D_w\right)=\int \int k\left(w_o,w_h\right)\iota \left(w_h,w_o,D_w\right){\mathit{\text{dw}}}_o{\mathit{\text{dw}}}_h$, where ι(w _h ,w _o ,D _w )=1 if w _h −w _o =D _w and ι=0 otherwise. The cumulative density function of D _w is noted F(D _w ).

Lemma 2.

where with probability F(D _R ) the migrant returns to the country of origin and enjoys E(g(c _o )|D _w ≤D _R ), where c _o depends on the realization of w _o . With probability (1−F(D _R )), the migrant stays in the host country and enjoys E(g(c _h )|D _w >D _R ). While the concavity of u(·) captures the migrant’s taste for consumption smoothing, the concavity of g(·) captures the degree of risk aversion. Section 4 and 5 are based on two particular cases of Kreps-Porteus preferences. Section 4 covers the case of risk-neutral migrants having a taste for consumption smoothing, which corresponds to a linear g(·) function and a concave u(·) function. This case allows us to introduce the “pure investment motive” for remittances, which appears when the expected return to remittances dominates that of savings. In Section 5, we introduce risk aversion by posing that u(·) and g(·) are identical. This second case, which corresponds to expected utility, puts forward the “precautionary motive” for remittances: depending on the distribution of wage risks across locations, remittances may decrease the variance of future consumption.

Proposition 1.

where $R_h^{\prime }\left(r\right)>0$ and $\frac{\mathrm{\partial E}\left(w_h\right|D_w>D_R)}{\mathrm{\partial r}}>0$.

Proof.

See Appendix 7. □

Remittances have an ambiguous impact on E(c _o |D _w ≤D _R ) because, ceteris paribus, migrants who remit more are ready to concede lower wages in the origin country. Indeed, as r increases, so does D _R , so that the condition to return (D _w ≤D _R ) is easier to satisfy. In other words, return migration is compatible with higher values of D _w , that is, lower (higher) values of w _o (w _h ). This result provides a rationale for poor economic outcomes of return migrants: the more migrants remit, the more likely they will return, and the lower the wages they are ready to accept in case of return. Conversely, remittances have a positive impact on E(w _h |D _w >D _R ): migrants who remit are less likely to stay, which implies that they will do so only for sufficiently large wages in the host country.

E[w] is the expected wage, based on the distributions of wages in each location conditional on the wage gap D _w . Similarly, E[R] is the total expectation of return to remittances, which also depends on ex-post location decisions. In order to show the impact of remittances on E[c₂], let us first analyze their impacts on E[w] and E[R] separately. The next Lemma shows that migrants who decide to remit anticipate lower expected wages, but higher expected benefits from remittances.

Lemma 3.

The first part of the lemma, which states that remittances have a negative impact on expected wages, is explained by the following intuition. Remitting increases the likelihood of return migration by $f\left(D_R\right)D_R^{\prime }$. The global distribution of wages is unaffected by this change, except at the margin where migrants are indifferent between returning and staying, i.e. when D _w =D _R . When switching from staying to returning, they renege on w _h to earn a wage w _o instead. Therefore, they suffer a loss D _w at this marginal point, which is precisely equal to D _R .

The second part of the lemma highlights two positive effects of r on E[R]. The first effect stems from the fact that, as for E[w], remittances increase the likelihood of return migration at the margin. At this margin, migrants enjoy a return R _o (r) instead of R _h (r): they benefit from a marginal increase equal to D _R . The second and most obvious term accounts for the marginal returns to remittances in each location, $R_o^{\prime }\left(r\right)$ and $R_h^{\prime }\left(r\right)$. Weighting these two marginal returns by their respective probabilities leads to E[R^′(r)].

Proposition 2.

Combining the two parts of Lemma 3 leads directly to Proposition 2. This proposition shows first that remittances affect E[c₂] through an increase in returns to remittances, but also a decrease in the expected wage, since remittances increase the likelihood of migrating back to the country of origin, where wages are on average lower. However, the net marginal effect of remittances on expected consumption is always positive, and corresponds to the expected marginal return to remittances, E[R^′(r)]. Figure 1 illustrates these effects. In this figure, when remittances are close to zero, the probability of return migration is zero, so that E[R(r)]=R _h (r), and E[w]=E[w _h ].¹⁹ As remittances increase, the likelihood of return migration becomes strictly positive, and E[R(r)] increases, as it becomes a convex combination of R _h (r) and R _o (r). Also, E[w] decreases, as it depends on the distribution of both w _h and w _o . As stated in the previous proposition, the sum of these two effects is always positive (the blue curve is always increasing in r). As remittances increase further, they become so large that the probability of returning to the origin country becomes equal to 1. In that case, E[R(r)]=R _o (r), and E[w]=E[w _o ]. Further increases in remittances no longer affect expected wages, while returns to remittances are perceived with certainty in the country of origin.

Secondly, the proposition shows that expected consumption is not only increasing, but may also be convex in r. This potential convexity comes from the first term in the right hand side of (12), which is positive and is due to the feedback effect of remittances. Indeed, even if R _o (r) and R _h (r) are not convex, expected consumption may be more and more increasing in remittances because the more the migrant remits, the higher the probability of returning to the origin country, where the benefits of remittances are the highest. In other words, as the migrant remits, E[R(r)] puts more weight on R _o (r), and less on R _h (r). In Figure 2, we present a case where R _o (r) and R _h (r) are both linear, so that, following the previous reasoning, returns to remittances are always convex: $\frac{{\partial }^{2}E\left[c_2\right]}{\partial r^2}=f\left(D_R\right){\left(D_R^{\prime }\left(r\right)\right)}^2\ge 0$. Figure 2 illustrates that as long as remittances increase the likelihood of return migration, E[c₂] is always convex in remittances when R _o (r) and R _o (r) are linear.

This possible variety in the “technology” of returns to remittances gives rise to the following questions: (i) which migrant characteristics are likely to lead to high/low or concave/convex returns to remittances, and (ii) how does this affect the migrant’s optimal remittances behavior? We address the first question in the next subsection, looking at the impact of migrants’ beliefs about their wage prospects.

Lemma 4.

For all r, the higher the migrant’s wage prospects, the lower $\frac{\mathrm{\partial E}\left[c_2\right]}{\mathrm{\partial r}}$.

Proof.

One can rewrite $\frac{\mathrm{\partial E}\left[c_2\right]}{\mathrm{\partial r}}=E\left[R^{\prime }\left(r\right)\right]$ as $F\left(D_R\left(r\right)\right)D_R^{\prime }+R_h^{\prime }\left(r\right)$, which clearly increases with F(D _R (r)). For a given level of D _R , a migrant with higher migration prospects faces a lower expected marginal return to remittances. See Appendix 7 for a graphical illustration. □

Let us now discuss how recipient characteristics affect $\frac{{\partial }^{2}E\left[c_2\right]}{\partial r^2}$. We have seen that $\frac{{\partial }^{2}E\left[c_2\right]}{\partial r^2}$ is the sum of two terms, $f\left(D_R\right){\left(D_R^{\prime }\left(r\right)\right)}^2$ and E[R^′′(r)]. While the first is always positive, the second term may also be positive, depending on the way recipients in the origin country use remittances. Indeed, E[R^′′(r)] is more likely to be positive in poor receiving households than in rich ones. Intuitively, poor families are likely to first allocate remittances to their basic needs. The more the migrant remits, the higher recipients’ capacity to switch to more productive investments, which also benefits the migrant. Marginal returns to remittances are therefore negligible for low levels of remittances, and become attractive as recipient households have improved their living conditions. As a result, migrants originating from poor households are more likely to face convex returns to remittances. This reasoning has been illustrated by Adams ([1998]), which shows that households with a migrant member have a higher marginal propensity to invest.

In the next section, we analyze migrants’ optimal saving and remittance decisions, taking into account the potential heterogeneity in the remittance technology across migrants.

Proposition 3.

Under risk neutrality, if $\frac{{\partial }^{2}E\left[c_2\right]}{\partial r^2}\le 0$, then three types of remittance-saving portfolios are possible: (0,s^∗), (r^∗,s^∗) and (r^∗,0), where:

under “high” migration prospects (large E[D _w ]): (0,s^∗) such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=\tau >E\left[R^{\prime }\left(0\right)\right]$,

under “intermediate” migration prospects: (r^∗,s^∗) such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=\tau =E\left[R^{\prime }\left(r\right)\right]$,

under “low” migration prospects (small E[D _w ]): (r^∗,0) such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=E\left[R^{\prime }\left(r^{\ast }\right)\right]>\tau$.

The marginal returns to both activities must equalize at this double interior solution. However, depending on the migrant’s wage prospects, one of these two first order conditions may not hold with equality, leading to a corner solution in either r or s. Following Lemma 4, if prospects are so high that E[R^′(0)]<τ, marginal returns to remittances are too low for the migrant to send remittances ; instead he/she will focus on savings. In contrast, if prospects are low, the optimal level of remittances may still have a marginal return which dominates that of savings: E[R^′(r^∗)]>τ. In this case, the migrant has no incentives to save and will only remit. These three cases are illustrated in Figure 3.

It must be noted that the solutions described here are valid if and only if $\frac{{\partial }^{2}E\left[c_2\right]}{\partial r^2}<0$. Intuitively, for the arbitrage condition, E[R^′(r)]=τ, to hold at equilibrium, remittances should have a decreasing expected marginal return. Indeed, remittances below the interior solution should have a higher marginal return than savings, while remittances beyond the interior solution should have a lower return than savings. If, as mentioned in the previous section, remittances have an increasing marginal return, this reasoning does not hold, and a corner solution will prevail. As shown in the next proposition, the case of convex returns to remittances is incompatible with an interior solution in both s and r.

Proposition 4.

If $\frac{{\partial }^{2}E\left[c_2\right]}{\partial r^2}>0$, then the optimal remittance/saving portfolio of the migrant is never diversified. Two remittance-savings portfolios are possible, (0,s^∗) and (r^∗,0), where:

under “high” migration prospects: (0,s^∗) with U(0,s^∗)>U(r^∗,0) and s^∗ such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=\tau$,

under “low” migration prospects: (r^∗,0) with U(r^∗,0)>U(0,s^∗), and r^∗ such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=E\left[R^{\prime }\left(r^{\ast }\right)\right]$.

Proof.

See Appendix 7. □

The main insight of this case is that the optimal remittance/saving portfolio of the migrant is never diversified when returns to remittances are convex. Which of the two assets is chosen depends again on return prospects. If prospects are high (low), marginal returns to remittances are low and the migrant saves (remits).

Figure 4 illustrates why an interior solution in both s and r is impossible under convex returns to remittances. On the left graph, for low values of r, marginal returns to remittances are too low compared to returns to savings, τ. Remitting large amounts in order to enjoy high marginal returns to remittances would distort the migrant’s consumption path excessively (insufficient consumption in period 1) compared to the balance offered by savings (point S). The graph in the middle represents the limit case in which the migrant is indifferent between only saving (point S) and only remitting (U(r^∗,0)=U(0,s^∗)). In this case, due to the convexity of returns to remittances, the migrant needs to remit a larger amount than what he/she would have saved (r^∗>s^∗) in order to reach the attractive portion of returns to remittances.²⁰ Finally, the last graph illustrates the case in which returns to remittances are large and/or very convex. Since marginal returns to remittances become quickly high, and the migrant only remits. Here, the migrant may consume more in period 1 than he/she would have with savings.

Let us now analyze the risk implications of remittances.

Lemma 5.

Proof.

See Appendix 7. □

Lemma 5 describes the effect of remittances on the variance of the migrant’s future consumption. The sign of this effect depends on two terms. First, remittances are likely to decrease V a r(c₂) if the expected consumption is lower in the origin country than in the host country (E(c _o |D _w ≤D _R )<E(c _h |D _w >D _R )). Note that these expectations are conditional on the wage gap, and that the distribution of wages across the two locations is in fact a crucial determinant of the sign of this first term. Second, remittances are likely to decrease V a r(c₂) if, at the margin where the migrant is indifferent between staying and returning (i.e. for D _w =D _R ), the variance of wages in the origin country is lower than the variance of wages in the host country.

In order to provide more insights from this general formula and to highlight the role played by the distribution of wage risks across locations, we need to add more structure on the distribution of wages. To this end, we will assume that wages are drawn from a bivariate normal distribution, with means E[w _o ]=μ _o and E[w _h ]=μ _h and variances $\mathit{\text{Var}}\left[w_o\right]={\sigma }_o^2$ and $\mathit{\text{Var}}\left[w_h\right]={\sigma }_h^2$. We assume without loss of generality that wages are independently distributed across locations, so that c o v(w _o ,w _h )=0.²¹ The reason why we assume a bivariate normal distribution is that this distribution has appealing properties allowing analytical solutions and providing intuitive results. One implication of normality is that the migrant’s income variance is minimized for E(c _o |D _w ≤D _R )=E(c _h |D _w >D _R ). This is intuitive since, at this point, uncertainty related to the location itself vanishes, given that ex ante the consumption levels are equal in both locations. It has to be noted, however, that if this level is optimal from an insurance perspective (precautionary motive), remittances are also potentially sent to increase the migrant’s expected income (pure investment motive). This combination of motives is explored in the section devoted to the migrant’s optimization problem.

Proposition 5.

Proof.

See Appendix 7. □

Proposition 5 indicates that remittances decrease the aggregate income risk faced by the migrant if her initial risk is mainly located in the host country (large θ _h ). Intuitively, increasing remittances makes return migration more likely, which reduces the exposure to the relatively large risk in the host country. An important implication of Proposition 5 is that ${\tilde{\theta }}_h$ is itself a function of the level of remittances. The next proposition states that the impact of remittances on risk is non-monotonic.

Proposition 6.

${\tilde{\theta }}_h$ is increasing in r. Therefore, V a r(c₂) is potentially non-monotonic in r, being decreasing for low r, and increasing for large r.

Proof.

See Appendix 7. □

This proposition states that low levels of remittances may decrease aggregate risk, as in this case ${\tilde{\theta }}_h$ is low and is likely to be smaller than θ _h . Large levels of remittances may on the other hand be risk-increasing, as ${\tilde{\theta }}_h$ is then high and will eventually be larger than θ _h .

It must be noted at this stage that the impact of remittances on aggregate risk is largely driven by the fact that our framework accounts for the location choice. More precisely, it can be argued that in this setting, the genuine self insurance device is the migrant’s ability to choose her location after information about wages is revealed. Indeed, for any initial geographical distribution of risks, the migrant is protected against low wages by the ability to select the best outcome among both locations. In other words, the migrant is capable of mitigating a negative shock in one location by moving to the other location. Moreover, as we show in the following example, this instrument is most powerful if risk is evenly distributed among locations. Example 1 provides the intuition behind propositions 5 and 6, namely that remittances are risk-reducing if risk is mainly located in the host economy and that transferring too much risk towards the origin country might, at the end, increase the aggregate risk faced by the migrant. In order to illustrate these points, we use a simplified framework where the only decision variable is location.

Example 1.

Proof.

See Appendix 7. □

As this simplified example illustrates, on the one hand, the location choice allows the migrant to increase expected consumption. On the other hand, it shows that aggregate risk depends on the initial risk composition. More precisely, the more the distribution of wage risks across locations is even ex ante, the lower the variance of consumption conditional on optimal location. If risk is mainly located in the host economy, remittances can be used to improve the balance of risks across locations, so that aggregate risk is reduced. This mechanism provides an explanation to the empirical evidence according to which migrants facing higher risks in the host country are more likely to remit (Amuedo-Dorantes and Pozo [2006]). On the contrary, as stated in Proposition 6, remittances may increase aggregate risk if the origin country is relatively riskier.

Definition 1.

Whether returns to remittances are convex or concave matters for the optimal migrant portfolio. Indeed, as in the section with risk neutrality, we know that the migrant’s objective may admit an interior maximum in both s and r only if total returns to remittances are concave. Although convex total returns to remittances are still possible for risk averse migrants, this possibility is less likely than in the case of risk neutral migrants. The fact that concavity is more relevant under risk aversion than under risk neutrality is due to the impact of remittances on the variance of consumption. By Proposition 6, we know that $\frac{{\partial }^2\mathit{\text{Var}}\left(c_2\right)}{\partial r^2}>0$. This effect, whose magnitude increases with the degree of absolute risk aversion η, increases the likelihood of concave returns to remittances. As a result, it is likely that risk averse migrants choose strictly positive levels of both savings and remittances. However, one should keep in mind that, even if returns to remittances are concave, the non-negativity constraint on savings may, under some conditions, be binding. As in Proposition 3, this will be the case if the return to savings is low compared to the returns to remittances, which, in the risk averse case, combine potential benefits in terms of both expectation and variance of consumption. For instance, because remittances are risk-reducing for illegal migrants, these migrants are more likely to have no savings, other things being equal.

Proposition 7.

“high” and “safe” migration prospects (large E[D _w ], low θ _h ): (0,s^∗) such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=\tau >\frac{\partial {\tilde{c}}_2}{\mathrm{\partial r}}{|}_{(r=0)}$,

“intermediate” migration prospects: (r^∗,s^∗) such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=\tau =\frac{\partial {\tilde{c}}_2}{\mathrm{\partial r}}{|}_{r^{\ast }}$,

“low” and “risky” migration prospects (low E[D _w ], large θ _h ): (r^∗,0) such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=\frac{\partial {\tilde{c}}_2}{\mathrm{\partial r}}{|}_{r^{\ast }}>\tau$.

“high” and “safe” migration prospects: (0,s^∗) with U(0,s^∗)>U(r^∗,0) and s^∗ such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=\tau$,

“low” and “risky” migration prospects: (r^∗,0) with U(r^∗,0)>U(0,s^∗) and r^∗ such that $\frac{u_1^{\prime }}{\delta u_2^{\prime }}=\frac{\partial {\tilde{c}}_2}{\mathrm{\partial r}}{|}_{r^{\ast }}$.

The last section provides a taxonomy of the motives for remittances, based on the analysis conducted above.