Following equation (4) and using a linear g(·) function, the riskneutral migrant’s utility is:
U=u\left({c}_{1}\right)+\mathrm{\delta u}\left(E\left[{c}_{2}\right]\right),
where c_{1} is defined in (1) and c_{2} is equal to c_{
o
} (2) if D_{
w
}<D_{
R
}, and is equal to c_{
h
} (3) otherwise. Since D_{
w
} is by definition equal to w_{
h
}−w_{
o
}, it belongs to the same data generating process. Therefore, for a given pair (w_{
o
},w_{
h
}) and its corresponding wage gap D_{
w
}, the following joint densities have the same values: k(w_{
o
},w_{
h
})=m(w_{
o
},D_{
w
})=n(w_{
h
},D_{
w
}), where m(w_{
o
},D_{
w
}) and n(w_{
h
},D_{
w
}) are joint densities of wages with D_{
w
}.^{17} Using these notations, we can write the expectation of period2 consumption by distinguishing the two final possible locations:
\phantom{\rule{20.0pt}{0ex}}E\left[{c}_{2}\right]\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\stackrel{{D}_{R}}{\underset{\infty}{\int}}\stackrel{+\infty}{\underset{\infty}{\int}}\phantom{\rule{0.3em}{0ex}}\left(\mathrm{\tau s}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{R}_{o}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{w}_{o}\right)\phantom{\rule{0.3em}{0ex}}m({w}_{o},{D}_{w}){\mathit{\text{dw}}}_{o}{\mathit{\text{dD}}}_{w}+\stackrel{+\infty}{\underset{{D}_{R}}{\int}}\stackrel{+\infty}{\underset{\infty}{\int}}\left(\mathrm{\tau s}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{R}_{h}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{w}_{h}\right)n({w}_{h},{D}_{w}){\mathit{\text{dw}}}_{h}{\mathit{\text{dD}}}_{w}.
(6)
Let us first analyze the properties of the remittance technology in the way it affects the expectation of future consumption. Based on these properties, we will study in a second subsection the optimal remittances decision of riskneutral migrants.
4.1 The impact of remittances on expected consumption
In this subsection, we will analyze the impact of remittances on (i) the expected consumption in each location, and (ii) the total expectation of consumption, taking into account optimal location decisions.
The first analysis will provide interesting results about the economic performance of return migrants and of permanent migrants. In order to study the expected consumption in each location conditional, let us rewrite (6) as:
E\left[{c}_{2}\right]=F\left({D}_{R}\right)E\left({c}_{o}\right{D}_{w}\le {D}_{R})+\left(1F\left({D}_{R}\right)\right)E({c}_{h}{D}_{w}>{D}_{R}),
(7)
where conditional expectations of consumption in case of return and permanent migration are respectively:
\begin{array}{cc}E\left({c}_{o}{D}_{w}\le {D}_{R}\right)& =\mathrm{\tau s}+{R}_{o}+E\left({w}_{o}{D}_{w}\le {D}_{R}\right),\\ E\left({c}_{h}{D}_{w}>{D}_{R}\right)& =\mathrm{\tau s}+{R}_{h}+E\left({w}_{h}{D}_{w}>{D}_{R}\right),\end{array}
and conditional expectations of wages are:
\begin{array}{cc}E\left({w}_{o}{D}_{w}\le {D}_{R}\right)& =\stackrel{{D}_{R}}{\underset{\infty}{\int}}\left(\stackrel{+\infty}{\underset{\infty}{\int}}{w}_{o}m\left({w}_{o}\right{D}_{w}){\mathit{\text{dw}}}_{o}\right)\frac{f\left({D}_{w}\right)}{F\left({D}_{R}\right)}{\mathit{\text{dD}}}_{w},\\ E\left({w}_{h}{D}_{w}>{D}_{R}\right)& =\stackrel{+\infty}{\underset{{D}_{R}}{\int}}\left(\stackrel{+\infty}{\underset{\infty}{\int}}{w}_{h}m\left({n}_{h}\right{D}_{w}){\mathit{\text{dw}}}_{h}\right)\frac{f\left({D}_{w}\right)}{1F\left({D}_{R}\right)}{\mathit{\text{dD}}}_{w}.\end{array}
This decomposition allows us to assess the impact of remittances on the expected consumption in each location separately, taking into account the endogeneity of the location decision. Such an exercise brings two interesting findings. First, it allows us to provide an explanation to the poor performance of some return migrants.^{18} Second, it shows that remittances, although they are more productive in case of return migration, increase the conditional expectation of consumption of migrants settling in the host country.
Proposition 1.
Conditionally on optimal location decisions in period 2,

1.
remittances decrease the expected wage E(w _{
o
}D _{
w
}≤D _{
R
}), but have an ambiguous impact on the expected consumption in the country of origin:
\frac{\mathrm{\partial E}\left({c}_{o}\right{D}_{w}\le {D}_{R})}{\mathrm{\partial r}}={R}_{o}^{\prime}\left(r\right)+\frac{\mathrm{\partial E}\left({w}_{o}\right{D}_{w}\le {D}_{R})}{\mathrm{\partial r}},
where {R}_{o}^{\prime}\left(r\right)\ge 0 but \frac{\mathrm{\partial E}\left({w}_{o}\right{D}_{w}\le {D}_{R})}{\mathrm{\partial r}}<0.

2.
remittances have a positive impact on the expected wage and consumption in the host country:
\frac{\mathrm{\partial E}\left({c}_{h}\right{D}_{w}>{D}_{R})}{\mathrm{\partial r}}={R}_{h}^{\prime}\left(r\right)+\frac{\mathrm{\partial E}\left({w}_{h}\right{D}_{w}>{D}_{R})}{\mathrm{\partial r}}>0,
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.
Having analyzed the impact of remittances by location, let us now analyze their global impact on E[c_{2}]. To this end, let us rewrite (6) as:
E\left[{c}_{2}\right]=\mathrm{\tau s}+E\left[R\right]+E\left[w\right],
(8)
where:
\begin{array}{ll}E\left[w\right]& \equiv F\left({D}_{R}\right)E\left({w}_{o}{D}_{w}\le {D}_{R}\right)+\left(1F\left({D}_{R}\right)\right)E\left({w}_{h}{D}_{w}>{D}_{R}\right),\phantom{\rule{2em}{0ex}}\end{array}
(9)
\begin{array}{ll}E\left[R\right]& \equiv F\left({D}_{R}\right){R}_{o}+\left(1F\left({D}_{R}\right)\right){R}_{h}.\phantom{\rule{2em}{0ex}}\end{array}
(10)
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 expost location decisions. In order to show the impact of remittances on E[c_{2}], 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.
Remittances have a negative impact on E[w] and a positive impact on E[R]:
\begin{array}{cc}\frac{\mathrm{\partial E}\left[w\right]}{\mathrm{\partial r}}& =f\left({D}_{R}\right){D}_{R}^{\prime}{D}_{R}<0,\\ \frac{\mathrm{\partial E}\left[R\right]}{\mathrm{\partial r}}& =f\left({D}_{R}\right){D}_{R}^{\prime}{D}_{R}+E\left[{R}^{\prime}\left(r\right)\right]>0,\end{array}
where
E\left[{R}^{\prime}\left(r\right)\right]\equiv F\left({D}_{R}\right){R}_{o}^{\prime}\left(r\right)+\left(1F\left({D}_{R}\right)\right){R}_{h}^{\prime}\left(r\right)>0.
(11)
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.
The total expectation of future consumption is increasing, and potentially convex in remittances.
\begin{array}{cc}\frac{\mathrm{\partial E}\left[{c}_{2}\right]}{\mathrm{\partial r}}& =\frac{\mathrm{\partial E}\left[R\right]}{\mathrm{\partial r}}+\frac{\mathrm{\partial E}\left[w\right]}{\mathrm{\partial r}}=E\left[{R}^{\prime}\left(r\right)\right]>0,\\ \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}+E\left[{R}^{\mathrm{\prime \prime}}\left(r\right)\right],\end{array}
(12)
where
E\left[{R}^{\mathrm{\prime \prime}}\left(r\right)\right]=F\left({D}_{R}\right){R}_{o}^{\mathrm{\prime \prime}}\left(r\right)+\left(1F\left({D}_{R}\right)\right){R}_{h}^{\mathrm{\prime \prime}}\left(r\right).
Combining the two parts of Lemma 3 leads directly to Proposition 2. This proposition shows first that remittances affect E[c_{2}] 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[c_{2}] 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.
4.2 How migrant characteristics affect the impact of r on E[c_{2}]
In this subsection, we discuss the impact of migrants’ and recipients’ characteristics on the shape of returns to remittances, both through \frac{\mathrm{\partial E}\left[{c}_{2}\right]}{\mathrm{\partial r}} and \frac{{\partial}^{2}E\left[{c}_{2}\right]}{\partial {r}^{2}}. While we provide here analytical results and their interpretation, the interested reader can find graphical representations of these results based on calibrations of the model in Appendix 7.
First, we show that the higher the migrants’ anticipations about their labor market performance in the host country, the lower their expected marginal returns to remittances. In order to show this, let us consider that some migrants have higher wage prospects from migration than others following the concept of first order stochastic dominance. A migrant with high (low) wage prospects faces a cumulative distribution of the wage gap between host and origin countries noted {F}_{{P}_{h}}\left({D}_{w}\right) ({F}_{{P}_{l}}\left({D}_{w}\right)). The distribution of the wage gap for migrants with high prospects firstorder stochastically dominates that of migrant with low wage prospects: {F}_{{P}_{h}}\left({D}_{w}\right)\le {F}_{{P}_{l}}\left({D}_{w}\right) for all D_{
w
}. In other words, for any given D_{
w
}, the higher the migrant’s wage prospects, the lower F(D_{
w
}). Note that first order stochastic dominance implies that migrants with higher prospects have a higher expected wage gap: {E}_{{P}_{h}}\left({D}_{w}\right)\ge {E}_{{P}_{l}}\left({D}_{w}\right).
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.
4.3 Optimal savings and remittances under risk neutrality
The riskneutral migrant’s objective is:
\underset{\left\{s,r\right\}}{\mathit{\text{Max}}}U=u\left({c}_{1}\right)+\mathrm{\delta u}\left(E\left[{c}_{2}\right]\right),
where c_{1} and E[c_{2}] are defined in (1) and (8), and both r and s need to satisfy nonnegativity constraints. While remittances can obviously not be negative, borrowing is precluded by the migrant’s lack of credibility to repay a loan in case of return migration. As mentioned in Proposition 2, returns to remittances may either be concave, or convex. Let us analyze these two cases separately.
Proposition 3.
Under risk neutrality, if \frac{{\partial}^{2}E\left[{c}_{2}\right]}{\partial {r}^{2}}\le 0, then three types of remittancesaving 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 migrant’s optimal strategy of saving/remittance portfolio is determined by the Kuhn Tucker conditions with respect to savings and remittances, which are respectively:
\begin{array}{cc}{U}_{s}& ={u}_{1}^{\prime}+\delta {u}_{2}^{\prime}\tau =0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}s>0,\\ ={u}_{1}^{\prime}+\delta {u}_{2}^{\prime}\tau <0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}s=0,\end{array}
(13)
and
\begin{array}{ll}{U}_{r}& ={u}_{1}^{\phantom{\rule{0.3em}{0ex}}\prime}+\delta {u}_{2}^{\prime}E\left[{R}^{\prime}\left(r\right)\right]=0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}r>0,\phantom{\rule{2em}{0ex}}\end{array}
(14)
\begin{array}{l}={u}_{1}^{\phantom{\rule{0.3em}{0ex}}\prime}+\delta {u}_{2}^{\prime}E\left[{R}^{\prime}\left(r\right)\right]<0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}r=0.\phantom{\rule{2em}{0ex}}\end{array}
(15)
If both first order conditions are satisfied with equality, that is, if there is an interior solution in both r and s, we obtain the following arbitrage condition:
E\left[{R}^{\prime}\left(r\right)\right]=\mathrm{\tau .}
(16)
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 remittancesavings 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.